Polatuzumab Vedotin, Venetoclax, and Rituximab and Hyaluronidase Human for the Treatment of Relapsed or Refractory Mantle Cell Lymphoma

Inclusion Criteria: * Age \>= 18 years * Pathologically confirmed relapsed or primary refractory mantle cell lymphoma with concurrent or prior tissue sample immunohistochemistry (IHC) positive for cyclin D1 or that is positive for fluorescence in situ hybridization (FISH) or cytogenetics for t(11;14) * NOTE: Safety Portion only: MCL or indolent B cell non-Hodgkin lymphoma (NHL), follicular lymphoma (FL) (grades I-IIIa), marginal-zone lymphoma (MZL) or small lymphocytic lymphoma (SLL) stratified as low risk for tumor lysis syndrome (TLS), relapsed or progressed after at least two lines of therapy, or one line of a BTK inhibitor containing therapy, or Autologous Stem Cell Transplant (AutoSCT). No limit to prior lines of therapy * NOTE: Expansion Portion only: MCL relapsed or progressed after at least two lines of therapy, or one line of a BTK inhibitor containing therapy, or Autologous Stem Cell Transplant (AutoSCT). No limit to number of prior therapies. May have received prior BTK inhibitor therapy * Measurable disease as defined with at least one lesion measuring \>= 1 x 1.5 cm by positron emission tomography (PET)-computed tomography (CT) using Lugano criteria * Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1000/mm\^3 (obtained =\< 14 days prior to registration) * Platelet count \>= 75,000/mm\^3 (obtained =\< 14 days prior to registration) * Hemoglobin \>= 9.0 g/dL (obtained =\< 14 days prior to registration) * International normalized ratio =\< 1.5 x upper limit of normal (ULN) for patients not receiving therapeutic anticoagulation (obtained =\< 14 days prior to registration) * Partial thromboplastin time (PTT) or activated PTT (aPTT) =\< 1.5 x upper limit of normal (ULN) (obtained =\< 14 days prior to registration) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x ULN (obtained =\< 14 days prior to registration) * Total bilirubin \< 1.5 x ULN (or =\< 3 x ULN for patients with documented Gilbert syndrome) (obtained =\< 14 days prior to registration) * Calculated creatinine (Cr) clearance \>= 45 ml/min using the modified Cockcroft-Gault formula (obtained =\< 14 days prior to registration) * Negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * NOTE: A female of childbearing potential is a sexually mature female who: * Has not undergone a hysterectomy or bilateral oophorectomy; or * Has not been naturally postmenopausal for at least 12 consecutive months (i.e. has had menses at any time in the preceding 12 consecutive months) * Able to provide informed written consent, and ability to comply with study related procedures * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * Willing to provide tissue samples for mandatory correlative research * For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use a contraceptive method with a failure rate of \< 1% per year during the treatment period and for at least 30 days after the last dose of venetoclax or 18 months after the last dose of rituximab and hyaluronidase human, whichever is longer. For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, and agreement to refrain from donating sperm, as defined below: * With female partners of childbearing potential, men must remain abstinent or use a condom plus an additional contraceptive method that together result in a failure rate of \< 1% per year during the treatment period and for at least 6 months after the last dose. Men must refrain from donating sperm during this same period. * With 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 to avoid exposing the embryo Exclusion Criteria: * Any of the following because this study involves an investigational agent whose genotoxic, mutagenic and teratogenic effects on the developing fetus and newborn are unknown: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational or chemotherapeutic agent which would be considered as a treatment for the primary neoplasm * Known CD20-negative status at relapse or progression * Prior allogeneic SCT * Completion of autologous SCT =\< 100 days prior to registration * Treatment with radioimmunoconjugate =\< 12 weeks prior to registration * Monoclonal antibody or antibody-drug conjugate (ADC) therapy within 5 half-lives or 4 weeks prior to registration, whichever is longer * Radiotherapy, chemotherapy, hormonal therapy, or targeted small-molecule therapy within 2 weeks prior to registration (with the exception of ibrutinib to prevent tumor flare, patients taking ibrutinib who are progressing must discontinue ibrutinib 2 half-lives or 2 days prior to initiating protocol therapy) * Clinically significant toxicity (other than alopecia) from prior therapy that has not resolved to grade =\< 2 (per National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\]5.0) prior to registration * Current grade \> 1 peripheral neuropathy * Any history of central nervous system (CNS) lymphoma or leptomeningeal infiltration * Treatment with systemic corticosteroids \> 20 mg/day prednisone or equivalent Patients who are receiving corticosteroids =\< 20 mg/day, prednisone or equivalent, for non-lymphoma treatment reasons must be documented to be on a stable dose for \>= 4 weeks prior to registration. If corticosteroid treatment is urgently required for lymphoma symptom control prior to the start of study treatment, up to 100 mg/day of prednisone or equivalent can be given for a maximum of 5 days, but all tumor assessments must be completed prior to start of corticosteroid treatment * History of severe allergic or anaphylactic reaction or known sensitivity to humanized or murine monoclonal antibodies rituximab, polatuzumab vedotin, or venetoclax * Active bacterial, viral, fungal, or other infection * Requirement for warfarin treatment (because of potential drug-drug interactions \[DDIs\] that may increase the exposure of warfarin) * Treatment with the following agents =\< 7 days prior to registration * Strong and moderate CYP3A inhibitors such as fluconazole, ketoconazole, and clarithromycin * Strong and moderate CYP3A inducers such as rifampin and carbamazepine. If subject is taking proton pump inhibitors, subject is willing to avoid co-administration and stagger venetoclax dosing * Consumption of grapefruit, grapefruit products, Seville oranges (including marmalade that contains Seville oranges), or star fruit =\< 3 days prior to registration * Clinically significant history of liver disease, including viral or other hepatitis, current alcohol abuse, or cirrhosis * Active hepatitis B or hepatitis C infection * NOTE: Patients who have been successfully treated and cleared their virus as evidenced by a negative hepatitis (Hep) B or Hep C polymerase chain reaction (PCR) are eligible * Known history of human immunodeficiency virus (HIV) positive status or known infection with human T-cell leukemia virus 1. For patients with unknown HIV status, HIV testing will be performed at screening * History of PML (progressive multifocal leukoencephalopathy) * Vaccination with a live virus vaccine =\< 28 days prior to registration * History of other malignancy that could affect compliance with the protocol or interpretation of results, with the exception of the following: curatively treated carcinoma in situ of the cervix, good-prognosis ductal carcinoma in situ of the breast, basal- or squamous-cell skin cancer, stage I melanoma, or low-grade, early-stage localized prostate cancer * Any previously treated malignancy that has been in remission without treatment for =\< 3 years prior to registration * Evidence of any significant, uncontrolled concomitant disease 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 previous 6 months, unstable arrhythmia, or unstable angina) or significant pulmonary disease (such as obstructive pulmonary disease or history of bronchospasm) * Major surgical procedure other than for diagnosis =\< 28 days prior to day 1 of cycle 1, or anticipation of a major surgical procedure during the course of the study * Inability or unwillingness to swallow pills * History of malabsorption syndrome or other condition that would interfere with enteral absorption * History of inflammatory bowel disease (e.g., Crohn's disease or ulcerative colitis) or active bowel inflammation (e.g., diverticulitis)