• 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