Polatuzumab Vedotin and Combination Chemotherapy With or Without Glofitamab for the Treatment of Untreated Aggressive Large B-cell Lymphoma

Inclusion Criteria:

* Untreated aggressive B-cell large-B cell lymphoma (non-Hodgkin lymphoma) with adverse features that may predict sub-optimal response to rituximab-cyclophosphamide, doxorubicin (hydroxydaunorubicin), vincristine (Oncovin), prednisone (R-CHOP) and in the opinion of the investigator would be treated with DA-EPOCH-R as standard of care. Subjects must be planned to receive full course (6 cycles) chemoimmunotherapy as per clinical standard of care. Composite lymphomas are not excluded provided that the subject has not receive prior systemic therapy for the indolent component and would receive DA-EPOCH-R as the standard of care regimen for the aggressive component. Eligible histologies based on 2016 World Health Organization (WHO) classification include:

  * High grade B-cell lymphoma with MYC and BCL2 and/or BCL6 translocations
  * High grade B-cell lymphoma, not otherwise specified (NOS)
  * Diffuse large B-cell lymphoma (DLBCL) NOS
  * Primary mediastinal B-cell lymphoma
  * T-cell/histiocyte-rich large-B-cell lymphoma
  * Epstein-Barr virus (EBV) + DLBCL, NOS
  * ALK+ large B-cell lymphoma (must be CD20+)
  * B-cell lymphoma, unclassifiable, with features intermediate between DLBCL and classical Hodgkin lymphoma
* Be willing and able to provide written informed consent for the trial
* Be \>= 18 years of age on day of signing informed consent
* Have measurable disease, including at least 1 nodal site measuring 1.5 cm or 1 extranodal site measuring 1.0 cm in longest dimension on computed tomography (CT) or fluorodeoxyglucose-positron emission tomography (FDG-PET)
* Have a performance status of 0-2 on the Eastern Cooperative Oncology Group (ECOG) performance scale (PS)
* Left ventricular ejection fraction (LVEF) \>= 50% on cardiac multiple-gated acquisition (MUGA) scan or cardiac echocardiogram (ECHO)
* Absolute neutrophil count (ANC) \>= 1,000/uL except in cases of marrow infiltration by lymphoma
* Platelets \>= 75,000 / mcL except in cases of marrow infiltration by lymphoma or hypersplenism
* Hemoglobin \>= 8 g/dL except in cases of marrow infiltration by lymphoma without red blood cell (RBC) transfusion within 14 days of first treatment
* Measured or calculated creatinine clearance (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]) \>= 40 mL/min.

  \* Creatinine clearance should be calculated per institutional standard
* Serum total bilirubin =\< 1.5 X upper limit of normal (ULN) (Patients with documented Gilbert disease may be enrolled if total bilirubin =\< 3.0 x ULN)
* Direct bilirubin =\< ULN for subjects with total bilirubin levels \> 1.5 ULN
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for subjects with liver involvement
* International Normalized Ratio (INR) or Prothrombin Time (PT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants
* Activated partial thromboplastin time (aPTT) =\< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants, or subject is shown to have an antiphospholipid antibody on workup
* For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of =\< 1% per year during the treatment period and for at least 12 months after the last dose of EPCH-R, 9 months after the last dose of polatuzumab, 2 months after the last dose of glofitamab (Arm B participants), or 3 months after the last dose of tocilizumab (if applicable), whichever is longer. Women must refrain from donating eggs during this same period. A woman is considered to be of childbearing potential if she is post-menarcheal, has not reached a postmenopausal state (=\<12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). The definition of childbearing potential may be adapted for alignment with local guidelines or requirements. Examples of contraceptive methods with a failure rate of =\<1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception
* For women of childbearing potential, a negative serum pregnancy test result during screening period. Women who are considered not to be of childbearing potential are not required to have a pregnancy test
* For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below: With female partners of childbearing potential or pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 6 months after the last dose of EPCH-R or polatuzumab, 2 months after the last dose of glofitamab (Arm B participants), or 2 months after the last dose of tocilizumab (if applicable), whichever is longer. Men must refrain from donating sperm during this same period. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or postovulation methods) and withdrawal are not acceptable methods of preventing drug exposure. Male patients considering preservation of fertility should bank sperm before study treatment
* ARM B ONLY: Negative SARS-CoV-2 antigen or PCR test within 7 days prior to enrollment

Exclusion Criteria:

* Contraindication to any of the individual components of glofitamab, tocilizumab or EPCH-R, including prior receipt of anthracyclines, or history of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibodies, or known sensitivity or allergy to murine products
* Prior systemic treatment for lymphoma with the exception of corticosteroids. Prior radiotherapy is allowed provided that this site is not used as a measurable site to assess response.
* Richter's transformation from chronic lymphocytic leukemia/small lymphocytic lymphoma is not allowed. Transformation from follicular or other indolent lymphomas is allowed provided that the subject has not received prior systemic therapy for their lymphoma and the aggressive component meets one of the criteria listed in inclusion criterion 1
* Diagnosis of Burkitt lymphoma
* Prior organ transplantation
* Current Grade \> 1 peripheral neuropathy by clinical examination or demyelinating form of Charcot-Marie-Tooth disease
* Prior systemic therapy for indolent lymphoma
* Prior use of any monoclonal antibody within 3 months of the start of cycle 1; any investigational therapy within 28 days prior to the start of cycle 1; vaccination with live vaccines within 28 days prior the start of cycle 1 and at any time during the study treatment period
* Prior therapy for large B-cell lymphoma except for patients who require lymphoma symptom control during screening may receive steroids in the following manner: Up to 30 mg/day of prednisone or equivalent may be used for lymphoma symptom control during screening, including prior to finalization of staging (not included as part of pre-phase treatment) If glucocorticoid treatment is urgently required at higher doses for lymphoma symptom control prior to the start of study treatment, tumor assessments must be completed prior to initiation of \> 30-100 mg/day of prednisone or equivalent. Prednisone \> 30-100 mg/day or equivalent may be given for a maximum of 7 days as a pre-phase treatment. If patients exceed the allowed dosing of corticosteroids, patients may still be eligible for the study provided that baseline imaging is performed (or repeated) after completion of the course of higher dose of steroids. Allowed corticosteroid dosing resets from the point of imaging forward and patients who do not exceed the allowed corticosteroid dosing from the point of imaging until initiation of study treatment may enroll
* History of other malignancy that could affect compliance with the protocol or interpretation of results except with permission of the principal investigator. The following are eligible without a specific waiver:

  * Patients with a history of curatively treated basal or squamous cell carcinoma or melanoma of the skin or in situ carcinoma of the cervix at any time prior to the study are eligible
  * Patients with any malignancy appropriately treated with curative intent and the malignancy has been in remission without treatment for \>= 2 years prior to enrollment are eligible
  * Patients with low-grade, early-stage prostate cancer (Gleason score 6 or below, Stage 1 or 2) with no requirement for therapy at any time prior to study are eligible
* Evidence of significant, uncontrolled, concomitant diseases that could affect compliance with the protocol or interpretation of results, including significant cardiovascular disease (such as New York Heart Association Class III or IV cardiac disease, myocardial infarction within the last 6 months, unstable arrhythmias, or unstable angina) or pulmonary disease (including obstructive pulmonary disease and history of bronchospasm)
* Recent major surgery (within 4 weeks prior to the start of cycle 1), other than for diagnosis
* History or presence of an abnormal electrocardiogram (ECG) that is clinically significant in the investigator's opinion, including complete left bundle branch block, second- or third-degree heart block, or evidence of prior myocardial infarction
* Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) which requires systemic treatment. Patients may proceed with screening during treatment for infection, but systemic treatment must be completed by cycle 1 day 1
* Positive test results for chronic hepatitis B infection (defined as positive hepatitis B surface antigen \[HBsAg\] serology):

  \* Patients with occult or prior hepatitis B infection (defined as positive total hepatitis B core antibody and negative HBsAg) may be included if hepatitis B virus (HBV) DNA is undetectable at the time of screening. These patients must be willing to undergo monthly DNA testing and appropriate antiviral therapy as indicated by institutional standard
* Positive test results for hepatitis C (hepatitis C virus \[HCV\] antibody serology testing)

  \* Patients positive for HCV antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA)
* History of uncontrolled human immunodeficiency virus (HIV)

  \* Patients with known diagnosis of HIV must have undetectable viral load, have a CD4 count ≥ 200/μL, and be on anti-retroviral therapy. HIV positive patients should be monitored per local/institutional standards while receiving study treatment
* Patients with a history of progressive multifocal leukoencephalopathy
* Pregnancy or lactation or intending to become pregnant during study
* ARM B ONLY: Known or suspected history of hemophagocytic lymphohistiocytosis (HLH)
* ARM B ONLY: Known or suspected chronic active Epstein-Barr virus infection
* ARM B ONLY: Current or past history of Waldenström macroglobulinemia
* ARM B ONLY: Current or past history of central nervous system (CNS) disease, such as stroke, epilepsy, CNS vasculitis, CNS lymphoma or neurodegenerative disease
* ARM B ONLY: Active autoimmune disease which is not well controlled by therapy
* ARM B ONLY: Participants with a history of autoimmune-related hypothyroidism on a stable dose of thyroid-replacement hormone may be eligible
* ARM B ONLY: Participants with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study
* ARM B ONLY: Participants with active autoimmune disease with dermatologic manifestations are eligible for the study
* ARM B ONLY: Participants with a history of autoimmune hepatitis, systemic lupus erythematosus, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener granulomatosis, Sjögren syndrome, multiple sclerosis, or glomerulonephritis will be excluded
* ARM B ONLY: Participants with a history of immune thrombocytopenic purpura, autoimmune hemolytic anemia, Guillain-Barré syndrome, myasthenia gravis, myositis, rheumatoid arthritis, vasculitis, or other autoimmune disease will be excluded unless they have not required systemic therapy in the last 12 months
* ARM B ONLY: Clinically significant liver disease, including active viral or other hepatitis, current alcohol abuse, or cirrhosis
* ARM B ONLY: Suspected active or latent tuberculosis (as confirmed by a positive interferon-gamma release assay)
* ARM B ONLY: Grade 3 infection within 4 weeks of treatment initiation