Inclusion
Documented informed consent of the participant and/or legally authorized representative
Assent, when appropriate, will be obtained per institutional guidelines
Agreement to allow the use of archival tissue from diagnostic tumor biopsies
If unavailable, exceptions may be granted with study principal investigator (PI) approval
Age: \>= 18 to 80 years
Eastern Cooperative Oncology Group =\< 2
Diagnosis of MCL established by histologic assessment including one of the following:
Immunohistochemistry of the biopsy
Flow cytometry of the biopsy
Evidence of t(11;14) translocation involving the cyclin D1 gene by fluorescence in situ hybridization (FISH), and/or cyclin D1 expression by immunohistochemistry (IHC) unless disease is morphologically consistent with MCL and has IHC expression of SOX11
Requiring treatment for MCL, and for which no prior systemic anticancer therapies have been received
Local radiotherapy not exceeding a total dose of 20 Gy at least 2 weeks prior the first dose of study therapy is allowed
Laboratory, radiographic, physical exam findings and/or symptoms attributable to MCL
Asymptomatic patients with blastoid or pleomorphic variant can be enrolled
High risk features as classified by Jain et al.
Blastoid/pleomorphic variants
Ki67 \>= 50%
Presence of a TP53 mutation defined by either molecular testing or IHC
del (17p) by FISH
Complex karyotype
High-risk Mantle Cell Lymphoma International Prognostic Index (MIPI) score (\>= 6.2)
Bulky disease
The presence of other high risk gene mutations (KMT2D, NSD2, NOTCH1, CDKN2A, NOTCH2, SMARCA4, CCND1) as long one of the other features above are present
Ability to swallow oral capsules/tablets
Without bone marrow involvement: absolute neutrophil count (ANC) \>= 1,000/mm\^3 With bone marrow involvement: ANC \>= 500/mm\^3
NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement
Without bone marrow involvement: Platelets \>= 75,000/mm\^3 With bone marrow involvement: Platelets \>= 25,000/mm\^3
NOTE: Platelet transfusions are not permitted within 14 days of platelet assessment unless cytopenia is secondary to disease involvement
Without bone marrow involvement: Hemoglobin \>= 8 g/dL With bone marrow involvement: Hemoglobin \>= 7 g/dL
NOTE: Red Blood cells transfusions are not permitted within 14 days of hemoglobin assessment unless cytopenia is secondary to disease involvement
Total bilirubin =\< 1.5 x upper limit of normal (ULN) (unless has Gilbert's disease)
Aspartate aminotransferase (AST) =\< 3.0 x ULN
Alanine aminotransferase (ALT) =\< 3.0 x ULN
Creatinine clearance of \>= 60 mL/min per 24 hour urine test or the Cockcroft-Gault formula
If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin (PT) =\< 1.5 x ULN If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants
If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =\< 1.5 x ULN If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants
Women of childbearing potential (WOCBP): negative serum pregnancy test
Agreement by females and males of childbearing potential to use an effective method of birth control (i.e., failure rate of \< 1% per year) or abstain from heterosexual activity for the course of the study treatment period through at least 30 days after the last dose of venetoclax and lenalidomide, 18 months after the last dose of obinutuzumab, 2 months after the last dose of glofitamab, or 4 months after the last dose of tocilizumab (if applicable) whichever is longer
Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only).
All study participants must be registered into the mandatory Revlimid Risk Evaluation and Mitigation Strategy (REMS) (registered trademark) program and be willing and able to comply with the requirements of the REMS (registered trademark) program (including use of aspirin \[ASA\]/ Food and Drug Administration \[FDA\] approved blood thinner)
Females of reproductive potential must adhere to the scheduled pregnancy testing as required in the Revlimid REMS (registered trademark) program
Exclusion
Treatment with the use of warfarin (because of potential drug-drug interactions that may potentially increase the exposure of warfarin)
Treatment with strong or moderate CYP3A inhibitors or strong CYP3A inducers within 4 weeks or 5 drug-elimination half-lives, whichever is longer, prior to initiation of study drug. Consumed grapefruit, grapefruit products, Seville oranges (including marmalade containing Seville oranges), or star fruit within 3 days prior to first dose of venetoclax
Ongoing corticosteroid use \> 30 mg/day of prednisone or equivalent. Patients who received corticosteroid treatment with =\< 30 mg/day of prednisone or equivalent must be documented to be on a stable dose of at least 4 weeks' duration prior to day 1 of cycle 1. Patients may have received a brief (=\< 7 days) course of systemic steroids (\>= 100 mg prednisone equivalent per day) prior to initiation of study therapy for control of lymphoma-related symptoms
Corticosteroid therapy for control of cancer symptoms is permitted
The use of inhaled corticosteroids is permitted
The use of mineralocorticoids for management of orthostatic hypotension is permitted
The use of physiologic doses of corticosteroids for management of adrenal insufficiency is permitted
Illicit drug or alcohol abuse within 12 months prior to screening, in the investigator's judgment
Prior solid organ transplantation within 60 months and requiring active immunosuppression
Receipt of live-virus vaccine within 28 days prior to the initiation of the study treatment or need for live-virus vaccines at any time during the study treatment
History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agents
History of other malignancy that could affect compliance with the protocol or interpretation of results
Patients with a history of curatively treated basal or squamous cell carcinoma of the skin or in situ carcinoma of the cervix are generally 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
Patients with any other malignancy appropriately treated with curative intent and the malignancy has been in remission without treatment for \>= 2 years prior to enrollment are eligible
Patients receiving adjuvant endocrine therapy for non-metastatic, hormone receptor positive breast cancer for \>= 2 years prior to enrollment are eligible
Major surgery (within 4 weeks prior to the start of the first dose of study treatment), other than for diagnosis
Known or suspected chronic active Epstein-Barr viral infection
Known or suspected history of hemophagocytic lymphohistiocytosis (HLH)
Known history of progressive multifocal leukoencephalopathy
Malabsorption syndrome or other condition that precludes enteral route of administration
Known allergy to both xanthine oxidase inhibitors and rasburicase
Positive for hepatitis C virus (HCV) virus by polymerase chain reaction (PCR) at screening. Testing only required if the hepatitis (Hep) C antibody is positive
Positive test results for hepatitis B virus (HBV) infection (defined as positive surface antigen \[HBsAg\]) at screening
Patients with occult or prior HBV infection (defined as negative HBsAg and positive hepatitis B core antibody \[HBcAb\]) may be included if HBV deoxyribonucleic acid (DNA) is undetectable, if they are willing to undergo DNA testing on day 1 of every cycle and every three months for at least 12 months after the last cycle of study treatment and appropriate antiviral therapy
Known active human immunodeficiency virus (HIV) infection. Subjects who have an undetectable or unquantifiable HIV viral load with CD4 \> 200 and are on highly active antiretroviral therapy (HAART) medication are allowed. Testing to be done only in patients suspected of having infections or exposures
Significant or extensive cardiovascular disease such has New York Heart Association class III or IV cardiac disease or Objective Assessment class C or D, myocardial infarction within the last 6 months, unstable arrythmias or unstable angina
Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) at study enrollment or any major episode of infection (as evaluated by the investigator) within 4 weeks prior to the first study treatment
Suspected or latent tuberculosis (confirmed by positive interferon -gamma release assay)
Females only: Pregnant or breastfeeding or intending to become pregnant during the study or within 18 months after the final dose of all study drugs
Women of childbearing potential must have a negative serum pregnancy test result within 14 days prior to initiation of study drug
History of uncontrolled autoimmune condition or disease requiring active systemic treatment to manage except when on a stable regimen for the treatment of hypothyroidism, Type 1 diabetes mellitus or psoriasis/eczema (topicals only)
Patients with history of immune thrombocytopenic purpura, autoimmune hemolytic anemia, Guillain-Barre 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
Prior treatment with systemic immunosuppressive medications (including, but not limited to, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor agents), within 4 weeks prior to first dose of study treatment
Patients with a history of autoimmune hepatitis, systemic lupus erythematosus, inflammatory bowel disease with active symptoms within last 60 months and on active immunosuppressive therapy, vascular thrombosis associated with antiphospholipid syndrome, Wegener granulomatosis, Sjogren syndrome, multiple sclerosis, or glomerulonephritis will be excluded
Primary or secondary central nervous system (CNS) lymphoma at the time of recruitment or history of CNS lymphoma
Current or history of CNS disease, such as stroke, epilepsy, CNS vasculitis, or neurodegenerative disease
Patients with a history of stroke who have not experienced a stroke or transient ischemic attack within the past 9 months and have no residual neurologic deficits, as judged by the investigator, are allowed
Clinically significant history of cirrhotic liver disease
Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures
Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)