Blinatumomab and Nivolumab With or Without Ipilimumab in Treating Patients With Poor-Risk Relapsed or Refractory CD19+ Precursor B-Lymphoblastic Leukemia

Inclusion Criteria:

* PRE-REGISTRATION ELIGIBILITY CRITERIA
* Patients must have suspected refractory or relapsed pre-B cell ALL or mixed phenotype acute leukemia (MPAL), or if newly diagnosed, the patient must be 60 years of age or older
* Bone marrow and/or peripheral blood specimens will be submitted for correlative studies; patients who have a dry tap will still be eligible
* REGISTRATION ELIGIBILITY CRITERIA
* Patients must have histologically or cytologically confirmed by the local institution CD19+ precursor B-acute lymphoblastic leukemia (pre-B cell ALL) OR CD19+ mixed phenotype acute leukemia (MPAL): a) with relapse following or refractory to at least one prior line of therapy if older than 21 years; b) in second or higher relapse or refractory to at least two prior lines of therapy if 21 years old and younger (16-21); c) or they must have a new diagnosis of pre-B cell ALL or CD19+ MPAL but are \>= 60 years old and are either not a candidate for or do not wish to receive traditional induction chemotherapy
* The evidence of CD19+ expression on leukemia cells must be confirmed by pathology review of the bone marrow and/or peripheral blood specimens (flow cytometry and/or immunohistochemistry) collected at the time of current relapse and prior to the initiation of therapy
* Patients with Philadelphia chromosome (Ph) positive (+) pre-B cell ALL OR Ph+ MPAL will be eligible if they have been refractory to or intolerant of treatment with at least 1 second-generation or third-generation tyrosine kinase inhibitor (TKI)
* Patients who were treated with blinatumomab in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells and did not experience unacceptable toxicities with prior blinatumomab administration; patients who were treated with chimeric antigen receptor (CAR)-modified T cells targeting CD19 in the past will be allowed on the study as long as they have persistent CD19 expression on leukemia cells
* Patients with a history of allogeneic hematopoietic stem cell transplantation (HSCT) will be eligible if they are more than 90 days removed from the date of stem cell infusion, have no evidence of acute graft-versus-host disease (GVHD) or active chronic (grade 2-4) GVHD, and are off of all transplant-related immunosuppression for at least 2 weeks
* Age \>= 16 years. Because no dosing or adverse event data are currently available on the use of blinatumomab in combination with nivolumab ± ipilimumab in pediatric patients, children are excluded from this study, but will be eligible for future pediatric trials.
* Eastern Cooperative Oncology Group (ECOG)/Karnofsky performance status of 0-2 (Karnofsky \>= 60%)
* Life expectancy of greater than 12 weeks
* Total bilirubin =\< 2.0 mg/dL (except patients with Gilbert syndrome, who can have total bilirubin \< 3.0 mg/dL)
* Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 5 x upper limit of normal (ULN)
* Serum creatinine =\< 1.5 x ULN OR creatinine clearance (CrCl) \>= 50 mL/min (if using the Cockcroft-Gault formula)
* The effects of nivolumab, ipilimumab, and blinatumomab on the developing human fetus are unknown; for this reason, women of child-bearing potential (WOCBP) and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; WOCBP should use an adequate method to avoid pregnancy for 23 weeks after the last dose of investigational drug; women of childbearing potential must have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 24 hours prior to the start of treatment on the study; women must not be breastfeeding; men who are sexually active with WOCBP must use any contraceptive method with a failure rate of less than 1% per year; men receiving nivolumab +/- ipilimumab and blinatumomab and who are sexually active with WOCBP will be instructed to adhere to contraception for a period of 31 weeks after the last dose of investigational product; women who are not of childbearing potential (i.e., who are postmenopausal or surgically sterile as well as azoospermic men) do not require contraception

  * Note: Women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal; menopause is defined clinically as 12 months of amenorrhea in a woman over 45 in the absence of other biological or physiological causes; in addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL
  * Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she (or the participating partner) should inform the treating physician immediately
* Adequate pulmonary function as assessed by oxygen saturation \>= 90% when ambulating and not requiring supplemental oxygen
* Patients with a known history of testing positive for human immunodeficiency virus (HIV) or known acquired immunodeficiency syndrome (AIDS) will be eligible if:

  * They are generally healthy from an HIV perspective and on a stable anti-retroviral regimen for \> 6 months
  * They have had no AIDS-defining conditions in the past 12 months other than historically low CD4+ cell counts
  * They have an undetectable viral load on standard assays
* Patients with a known history of hepatitis C (HCV) will be eligible if they have an undetectable viral load; if the patient received treatment for HCV, then that treatment must have been completed at least three weeks prior to enrollment
* Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

* Patients who have had chemotherapy or other systemic therapy or radiotherapy, or those who have not recovered from adverse events due to prior administered agents as follows: chemotherapy, radiotherapy or surgery =\< 3 weeks prior to entering the study, targeted therapy (e.g., TKI) =\< 1 week prior to entering the study; autologous HSCT =\< 6 weeks prior to entering the study; investigational drug or immunotherapy (e.g. rituximab) =\< 4 weeks prior to entering the study; prophylactic intrathecal chemotherapy within one week of enrollment allowed; patients will be allowed to receive cytoreduction with hydroxyurea, 6-mercaptopurine, corticosteroids (dexamethasone, prednisone or similar) or cyclophosphamide provided that it is discontinued at least 24 hours prior to the initiation of study treatment; pre-phase treatment with dexamethasone 10 mg/m\^2 (maximum total 24 mg per day) for up to 5 days is required for patients with bone marrow blasts more than 50%, peripheral blood blasts of 15,000/uL or higher, or elevated lactate dehydrogenase suggesting rapidly progressing disease as per investigator's assessment; pre-phase treatment must be stopped at least 24 hours prior to the initiation of blinatumomab
* Patients who are receiving any other investigational agents
* Patients should be excluded if they have had prior treatment with an anti-PD-1, anti-PD-L1, anti-PD-L2, anti-CTLA-4 antibody, or any other antibody or drug specifically targeting T-cell co-stimulation or immune checkpoint pathways
* Patients with active central nervous system leukemia are excluded from this clinical trial because they may develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; patients with a history of central nervous system (CNS) leukemia but no active disease at the time of enrollment are eligible; the absence of CNS disease must be confirmed by flow cytometric and cytologic examination of the cerebrospinal fluid (CSF) within 7 days of study enrollment
* Active leukemia in the testes or isolated extramedullary relapse; patients with a history of treated leukemia in testes but no active disease at the time of enrollment are eligible
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to nivolumab, ipilimumab, or blinatumomab
* History of severe hypersensitivity reaction to any monoclonal antibody
* Uncontrolled intercurrent illness including, but not limited to, active, uncontrolled infection; symptomatic congestive heart failure; unstable angina pectoris; cardiac arrhythmia; or psychiatric illness/social situations that would limit compliance with study requirements; patients with infection under treatment and controlled with antibiotics are eligible
* Pregnant women are excluded from this study because nivolumab and ipilimumab have the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with nivolumab and ipilimumab, breastfeeding should be discontinued if the mother is treated with nivolumab and ipilimumab; these potential risks may also apply to blinatumomab
* History of any chronic hepatitis including alcoholic, non-alcoholic steatohepatitis (NASH), drug related, autoimmune, chronic viral positive tests for hepatitis B virus surface antigen (HBsAg), hepatitis B core antibody (anti-HBc) in the absence of hepatitis B surface antibody (anti-HBs), or a positive hepatitis C (HCV) viral load; these patients are excluded due to the risk for autoimmune hepatitis with immune checkpoint inhibitors exacerbating their known liver disease as well as the unknown risk for hepatitis B and/or C reactivation with blinatumomab and immune checkpoint inhibitors
* Subjects with active autoimmune disease, a history of known or suspected autoimmune disease or a history of a syndrome requiring systemic corticosteroids (\> 10 mg daily of prednisone equivalent) except for the treatment of malignancy with the exception of:

  * Isolated vitiligo
  * Resolved childhood atopy
  * History of a positive antinuclear antibody (ANA) titer without associated symptoms or history of symptoms of an autoimmune disorder
  * Controlled thyroid disorders
  * Type I diabetes mellitus
  * Psoriasis, Sjogren's syndrome, and arthropathies not requiring systemic treatment
  * Autoimmune diseases: these include but are not limited to patients with a history of immune related neurologic disease, multiple sclerosis, autoimmune (demyelinating) neuropathy, Guillain-Barre syndrome, myasthenia gravis; systemic autoimmune disease such as systemic lupus erythematosus (SLE), connective tissue diseases, scleroderma, autoimmune vasculitis, inflammatory bowel disease (IBD), Crohn's, ulcerative colitis, hepatitis; and patients with a history of toxic epidermal necrolysis (TEN), Stevens-Johnson syndrome, or phospholipid syndrome should be excluded because of the risk of recurrence or exacerbation of disease
* Patients should be excluded if they have a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days of study drug administration; inhaled or topical steroids and adrenal replacement doses =\< 10 mg daily prednisone equivalents are permitted in the absence of active autoimmune disease; patients are permitted to use topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption); physiologic replacement doses of systemic corticosteroids are permitted, even if \> 10 mg/day prednisone equivalents; a brief course of corticosteroids for prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions (e.g., delayed-type hypersensitivity reaction caused by contact allergen), or as pre-phase treatment for cytoreduction; patients will receive steroids with blinatumomab to reduce cytokine release syndrome (CRS) as specified in the protocol
* Patients who have had evidence of active or acute diverticulitis, intra-abdominal abscess, gastrointestinal (GI) obstruction and abdominal carcinomatosis which are known risk factors for bowel perforation should be evaluated for the potential need for additional treatment before coming on study
* Patients who have a history of clinically relevant CNS pathology such as epilepsy, seizure, paresis, aphasia, stroke, severe brain injuries, dementia, Parkinson's disease, cerebellar disease, psychosis, or other significant CNS abnormalities; a history of treated CNS leukemia will be allowed if recent CNS studies confirm the absence of active CNS disease at the time of study entry (screening)
* Patients with a known concurrent malignancy that is progressing or requires active treatment; exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy or carcinoma in situ of the cervix
* Subjects with interstitial lung disease that is symptomatic or may interfere with the detection or management of suspected drug-related pulmonary toxicity