Testing the Addition of a New Anti-cancer Drug, Venetoclax, to the Usual Treatment (Ibrutinib and Obinutuzumab) in Untreated, Older Patients With Chronic Lymphocytic Leukemia

Inclusion Criteria:

* PRE-REGISTRATION ELIGIBILITY CRITERIA (STEP 0)
* Patients must have been diagnosed with chronic lymphocytic leukemia (CLL) (\> 5000 B-cells per uL of peripheral blood at any point during the course of their disease) or small lymphocytic lymphoma (SLL) with \< 5000 B-cells per uL of blood but with disease-associated lymphadenopathy
* This blood submission is mandatory prior to registration/randomization to perform fluorescence in situ hybridization (FISH) centrally that will be used for stratification. It should be obtained as soon after pre-registration as possible
* REGISTRATION ELIGIBILITY CRITERIA (STEP 1)
* Patients must be diagnosed with CLL or SLL in accordance with 2018 International Workshop on Chronic Lymphocytic Leukemia (IWCLL) criteria that includes all of the following:

  * \>= 5 x10\^9 B lymphocytes (5000/uL) in the peripheral blood measured by flow cytometry at any point in the course of the disease or less peripheral blood involvement but disease-associated lymphadenopathy
  * On local morphologic review, the leukemic cells must be small mature lymphocytes, and prolymphocytes must not exceed 55% of the blood lymphocytes
  * Neoplastic cells on immunophenotype (performed locally) must reveal a clonal B-cell population, which express the B cell surface markers of CD19 and CD20, as well as the T-cell antigen CD5. Patients with bright surface immunoglobulin expression or lack of CD23 expression in \>10% of cells must lack t(11;14) translocation by interphase cytogenetics
* Patients must be intermediate or high-risk Rai stage CLL or SLL

  * Intermediate risk (formerly stage I/II) is defined by lymphadenopathy and/or hepatomegaly or splenomegaly without anemia or thrombocytopenia
  * High risk (formerly stage III/IV) is defined by splenomegaly and/or anemia (hemoglobin \< 11g/dL) not attributable to autoimmune hemolytic anemia and/or thrombocytopenia (platelets \[plt\] \< 100 x10\^9/L) not attributable to autoimmune thrombocytopenia
* Patients must meet criteria for treatment as defined by 2018 IWCLL guidelines which includes at least one of the following criteria:

  * Evidence of marrow failure as manifested by the development or worsening of anemia or thrombocytopenia (not attributable to autoimmune hemolytic anemia or thrombocytopenia)
  * Massive (\>= 6 cm below the costal margin), progressive or symptomatic splenomegaly
  * Massive nodes (\>= 10 cm) or progressive or symptomatic lymphadenopathy
  * Progressive lymphocytosis with a lymphocyte doubling time \< 6 months or an increase of \>= 50% over a 2 month period
  * Autoimmune anemia and/or thrombocytopenia that is poorly responsive to standard therapy
  * Symptomatic or functional extranodal involvement (e.g. skin, kidney, lung, spine)
  * Constitutional symptoms, which include any of the following:

    * Unintentional weight loss of 10% or more within 6 months
    * Significant fatigue
    * Fevers \> 100.5 degrees Fahrenheit (F) for 2 weeks or more without evidence of infection
    * Night sweats \>= 1 month without evidence of infection
* Treatment with rituximab and/or high dose corticosteroids for autoimmune complications of CLL/SLL must be complete at least 4 weeks prior to enrollment. Palliative steroids must be at a dose not higher than 20 mg/day of prednisone or equivalent corticosteroid at the time of registration
* Age \>= 65 years
* Eastern Cooperative Oncology Group (ECOG) performance status 0-2
* Absolute neutrophil count (ANC) \>= 1,000/mm\^3 except if due to bone marrow involvement
* Platelet count (untransfused) \>= 30,000/mm\^3 except if due to bone marrow involvement
* Calculated (Calc.) creatinine clearance \>= 40 mL/min (by Cockcroft-Gault)
* Bilirubin =\< 1.5 x upper limit of normal (ULN) except if due to liver involvement, hemolysis, or Gilbert's disease
* Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 x upper limit of normal (ULN) except if due to liver involvement
* If evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be undetectable on suppressive therapy if indicated
* Please note: Intravenous immunoglobulin therapy (IVIG) can cause a false positive hepatitis B serology. If patients receiving routine IVIG have core antibody or surface antigen positivity without evidence of active viremia (negative hepatitis B deoxyribonucleic acid \[DNA\]) they may still participate in the study, must have hepatitis serologies and hepatitis B DNA monitored periodically by the treating physician
* If history of hepatitis C virus (HCV) infection, must be treated with undetectable HCV viral load
* Human immunodeficiency virus (HIV)-infected patients on effective antiretroviral therapy with undetectable viral load within 6 months are eligible for this trial
* Central fluorescent in situ hybridization (FISH) blood results are mandatory prior to registration/randomization for it will be used for stratification
* Patients must be able to swallow capsules and not have the following conditions: disease significantly affecting gastrointestinal absorption, resection of the stomach or small bowel, partial or complete bowel obstruction
* Patients must be able to receive either a xanthine oxidase inhibitor or rasburicase for prophylaxis/treatment of tumor lysis syndrome (TLS)
* RE-REGISTRATION ELIGIBILITY CRITERIA (STEP 2)
* Completion of treatment through cycle 14 day 28, and remain on ibrutinib therapy
* Receipt of central BM MRD results
* Response assessment completed with CR determination

Exclusion Criteria:

* Patients must not have had prior therapy for CLL/SLL (except palliative steroids or treatment of autoimmune complications of CLL with rituximab or steroids)
* Patients must not have any history of Richter's transformation or prolymphocytic leukemia (prolymphocytes in blood \> 55%)
* Patients with class III or class IV heart failure by New York Heart Association, those with unstable angina, and those with uncontrolled arrhythmia are not eligible
* Patients who have had a myocardial infarction, intracranial bleed, or stroke within the past 6 months are not eligible
* Patients must not be receiving active systemic anticoagulation with heparin or warfarin. Patients on warfarin must discontinue the drug for at least 10 days prior to registration on the study
* Chronic concomitant treatment with strong inhibitors of CYP3A4/5 is not allowed on this study. Patients on strong CYP3A inhibitors must discontinue the drug for 14 days prior to registration on the study
* Chronic concomitant treatment with strong CYP3A4/5 inducers is not allowed. Patients must discontinue the drug 14 days prior to registration on the study
* Patients must not require more than 20 mg prednisone or equivalent corticosteroid daily
* Patients must not have uncontrolled active systemic infection requiring intravenous antibiotics
* Patients must not have a known allergy to mannitol
* Patients must not have prior significant hypersensitivity to rituximab (not including infusion reactions)
* Patients may not have had major surgery within 10 days prior to registration, or minor surgery within 7 days prior to registration. Examples of minor surgery include dental surgery, insertion of a venous access device, skin biopsy, or aspiration for a joint. The decision about whether a surgery is major or minor can be made at the discretion of the treating physician