Sonrotoclax, Rituximab, and Zanubrutinib in Treating Participants With Chronic Lymphocytic Leukemia, Small Lymphocytic Lymphoma, and Mantle Cell Lymphoma

Inclusion Criteria: * Provision of signed and dated written informed consent prior to any study-specific procedures, sampling, or analyses * Age 18 years or older * Confirmed diagnosis (per World Health Organization \[WHO\] guidelines, unless otherwise noted) of one of the following: * CLL/SLL COHORT: CLL/SLL diagnosis that meets the International Workshop on Chronic Lymphocytic Leukemia criteria: * Meeting the following sets of prior treatment criteria: * For the R/R cohort, disease that relapsed after, or was refractory to, at least 1 prior therapy * For the treatment-naïve cohort, patients should have no prior treatment for CLL/SLL (other than 1 aborted regimen \< 2 weeks in duration and \> 4 weeks before enrollment) * Requiring treatment per International Workshop on CLL (iwCLL) criteria * MCL COHORT: WHO-defined MCL * R/R MCL is defined as a disease that relapsed after, or was refractory to, at least 1 prior systemic therapy * Measurable disease, defined as: * CLL/SLL: at least 1 lymph node \> 1.5 cm in longest diameter and measurable in 2 perpendicular dimensions by computed tomography (CT)/magnetic resonance imaging (MRI) or clonal lymphocytes \>= 5 x 109/L present on peripheral blood flow cytometry * MCL, or SLL: at least 1 lymph node \> 1.5 cm in the longest diameter OR 1 extranodal lesion \> 1.0 cm in the longest diameter, measurable in 2 perpendicular dimensions by CT/MRI * Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2 * Absolute neutrophil count (ANC) \>= 1.0 x 10\^9/L =\< 7 days before the first dose of the study drug with or without growth factor support. There is an exception for patients with bone marrow involvement, in which case ANC must be \>= 0.75 x 10\^9/L before the first dose of the study drug * Platelets \> 75,000 x 10\^9/L (\> 75,000 cells/mm\^3) =\< 7 days before the first dose of the study drug without the use of growth factor support or platelet transfusions. Patients with bone marrow involvement will be allowed to have a platelet count \> 50,000 x 10\^9/L (\> 50,000 cells/mm\^3) =\< 7 days before the first dose of the study drug without the use of growth factor support or platelet transfusions * Hemoglobin \> 75 g/L =\< 7 days before the first dose of the study drug (with or without transfusion) * Creatinine clearance or glomerular filtration rate (GFR) \>= 50 mL/min as estimated by one of the following: * Cockcroft-Gault equation * Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation * 24-hour urine collection * Aspartate aminotransferase (AST)/serum glutamic-oxaloacetic transaminase =\< 2 x upper limit of normal (ULN) * Alanine aminotransferase (ALT)/serum glutamic-pyruvic transaminase =\< 2 x ULN * Total bilirubin level =\< 1.5 x ULN (unless documented Gilbert's syndrome). For patients with documented Gilbert's syndrome, total bilirubin may exceed this value, but direct bilirubin must be =\< 1.0 x ULN * Serum amylase =\< 1.5 x ULN * Serum lipase =\< 1.5 x ULN * Women of childbearing potential (WOCBP) must have a negative serum pregnancy test =\< 7 days before the first dose of the study drug. In addition, they must use a highly effective method of birth control initiated before the first dose of the study drug, for the duration of the study treatment period, and for \>= 180 days after the last dose of the study drug * NOTE: WOCBP is a woman who: 1) has achieved menarche at some point, 2) has not undergone a hysterectomy or bilateral oophorectomy or 3) has not been naturally postmenopausal (amenorrhea following cancer therapy does not rule out childbearing potential) for at least 24 consecutive months (ie, has had menses at any time in the preceding 24 consecutive months) * NOTE: Highly effective contraceptive methods include the following: * Combined (estrogen and progestogen-containing) hormonal contraception associated with the inhibition of ovulation. Combined hormonal contraception may be oral, intravaginal, or transdermal * Progestogen-only hormonal contraception associated with the inhibition of ovulation. Progesterone-only hormonal contraception may be oral, injectable, or implantable * An intrauterine device * Intrauterine hormone-releasing system * Bilateral tubal * Vasectomized partner * Sexual abstinence (defined as refraining from heterosexual intercourse during the entire period of risk associated with the study treatment, starting the day before the first dose of study treatment, for the duration of the study, and for \>= 180 days after the last dose of study drug. Total sexual abstinence should only be used as a contraceptive method if it is in line with the patients' usual and preferred lifestyle. Periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods), declaration of abstinence for the duration of exposure to investigational medicinal product, and withdrawal are not acceptable methods of contraception * Of note, barrier contraception (including male and female condoms with or without spermicide) is not considered a highly effective method of contraception, and, if used, this method must be used in combination with another acceptable method listed above * For patients using hormonal contraceptives such as birth control pills or devices, a second barrier method of contraception (e.g., condoms) must be used * Nonsterile men must use a highly effective method of birth control along with barrier contraception for the duration of the study treatment period and for ≥ 180 days after the last dose of the study drug. During this same period, they must not donate sperm. Sterile men must use barrier contraception * Life expectancy of \> 6 months * Able to comply with the requirements of the study Exclusion Criteria: * Exposure to a Bcl-2 inhibitor within the last 12 months or a history of disease progression while taking a Bcl-2 inhibitor * Prior malignancy (other than the disease under study) within the past 2 years, except for curatively treated basal or squamous skin cancer, melanoma, superficial bladder cancer, carcinoma in situ of the cervix or breast, or localized Gleason score =\< 6 prostate cancer * Underlying medical conditions that may render the administration of study drug hazardous or obscure the interpretation of safety or efficacy results * Known current central nervous system involvement by lymphoma/leukemia * Known plasma cell neoplasm other than a monoclonal gammopathy of undetermined significance (MGUS), prolymphocytic leukemia, or history of or currently suspected Richter's syndrome * Prior autologous stem cell transplant unless \>= 3 months after transplant; or prior chimeric antigen receptor T-cell (CAR-T) therapy unless \>= 3 months after cell infusion * Prior allogeneic stem cell transplant with active graft-versus-host disease (GVHD), or requiring immunosuppressive drugs for the treatment of GVHD, or have taken calcineurin inhibitors within 4 weeks prior to consent * History of a severe bleeding disorder such as hemophilia A, hemophilia B, von Willebrand disease, or history of spontaneous bleeding requiring blood transfusion or other medical intervention * Use of the following substances prior to the first dose of the study drug: * =\< 28 days before the first dose of the study drug: * Any biologic and/or immunologic-based therapy(ies) including experimental therapy(ies) for leukemia, lymphoma, or myeloma (including, but not limited to, monoclonal antibody therapy, e.g., rituximab, and/or cancer vaccine therapy). If biological and/or immunologic-based therapy(ies) are used for non-oncological indications, enrollment will be at the discretion of the principal investigator (PI) * =\< 14 days before the first dose of the study drug: * Systemic chemotherapy or radiation therapy * =\< 7 days before the first dose of the study drug: * Corticosteroid given with antineoplastic intent * =\< 3 days (or 5 half-lives; whichever is shorter) before the first dose of the study drug: * Bruton's tyrosine kinase inhibitor (BTKi) or other small molecule inhibitor is given with antineoplastic intent * Active fungal, bacterial, and/or viral infection requiring systemic therapy * Note: oral antibiotics for minor bacterial infections are allowed * Major surgery =\< 4 weeks before the first dose of study treatment * Toxicity from prior anticancer therapy that has not recovered to grade =\< 1 (except for alopecia, ANC, and platelet count; for ANC and platelet count) * Clinically significant cardiovascular disease including the following: * Myocardial infarction =\< 6 months before screening * Unstable angina =\< 3 months before screening * New York Heart Association class III or IV congestive heart failure * History of clinically significant arrhythmias (e.g., sustained ventricular tachycardia, ventricular fibrillation, torsades de pointes) * Heart rate-corrected QT interval \> 480 milliseconds based on Fridericia's formula * History of Mobitz II second-degree or third-degree heart block without a permanent pacemaker in place * Uncontrolled hypertension as indicated by a minimum of 2 consecutive blood pressure measurements, at screening, showing systolic blood pressure \> 170 mmHg and diastolic blood pressure \> 105 mmHg * Known infection with human immunodeficiency virus (HIV) or serologic status reflecting active viral hepatitis B (HBV) or viral hepatitis C (HCV) infection as follows: * Presence of viral hepatitis B surface antigen (HBsAg) or viral hepatitis B core antibody (HBcAb) * Note: Patients with the presence of HBcAb, but absence of HBsAg, are eligible if HBV deoxyribonucleic acid (DNA) is undetectable and if they are willing to take HBV reactivation prophylaxis and undergo monitoring for HBV reactivation * Presence of HCV antibody * Note: Patients with the presence of HCV antibody are eligible if HCV ribonucleic acid (RNA) is undetectable and if they are willing to undergo monitoring for HCV reactivation * Pregnant or lactating women * Unable to swallow capsules or disease significantly affecting gastrointestinal function such as malabsorption syndrome, resection of the stomach or small bowel, bariatric surgery procedure, symptomatic inflammatory bowel disease, or partial or complete bowel obstruction * Inability to comply with study procedures * Receiving any treatment with a strong or moderate CYP3A4 inhibitor =\< 14 days (or 5 half-lives, whichever is longer) before the first dose of sonrotoclax * Unwillingness to stop consumption of grapefruit, grapefruit products, Seville oranges, or starfruit within 3 days before the first dose of sonrotoclax or during the study * Receiving any treatment with a strong CYP3A4 inducer =\< 14 days (or 5 half-lives, whichever is longer) before first dose of sonrotoclax * History of interstitial lung disease, noninfectious pneumonitis, or uncontrolled lung diseases, including but not limited to pulmonary fibrosis and acute lung diseases * Autoimmune anemia and/or thrombocytopenia that is poorly controlled by corticosteroids or other standard therapy * Ongoing, drug-induced liver injury, alcoholic liver disease, nonalcoholic steatohepatitis, primary biliary cirrhosis, ongoing extrahepatic obstruction caused by cholelithiasis, cirrhosis of the liver, or portal hypertension * Receiving drugs known to prolong the QT/corrected QT (QTc) interval * Vaccination with a live vaccine =\< 35 days before the first dose of the study drug * Note: Seasonal vaccines for influenza are generally inactivated vaccines and are allowed. Intranasal vaccines are live vaccines and are not allowed