Tafasitamab and Zanubrutinib for the Treatment of Patients With Newly Diagnosed Chronic Lymphocytic Leukemia or Small Lymphocytic Lymphoma, TaZA CLL Study

Inclusion Criteria: * Documented informed consent of the participant and/or legally authorized representative * Assent, when appropriate, will be obtained per institutional guidelines * Age: \>= 18 years * Eastern Cooperative Oncology Group (ECOG) =\< 2 * Histologically or flow cytometry confirmed diagnosis of B-CLL/SLL as documented by medical records and with histology based on criteria established by the World Health Organization (WHO) * No prior treatment for CLL, except steroids and/or rituximab to treat autoimmune complications * Active disease meeting criteria for requiring treatment per the iwCLL 2018 guidelines * A minimum of any one of the following constitutional symptoms: * Unintentional weight loss \> 10% within the previous 6 months prior to screening * Extreme fatigue (unable to work or perform usual activities) * Fevers of greater than 100.5 degrees Fahrenheit (F) for \>= 2 weeks without evidence of infection * Night sweats without evidence of infection * Evidence of progressive marrow failure as manifested by the development of, or worsening of anemia or thrombocytopenia * Massive (i.e., \> 6 cm below the left costal margin), progressive or symptomatic splenomegaly * Massive nodes or clusters (i.e., \> 10 cm in longest diameter) or progressive lymphadenopathy * Progressive lymphocytosis with an increase of \> 50% over a 2-month period, or an anticipated doubling time of less than 6 months * Autoimmune anemia or thrombocytopenia that is poorly responsive to corticosteroids * Symptomatic or functional extranodal involvement (eg, skin, kidney, lung, spine) * Participant must be able to swallow tablets or capsules. A participant with any gastrointestinal disease that would impair ability to swallow, retain, or absorb drug is not eligible * Absolute neutrophil count (ANC) \>= 1,000/mm\^3 unless due to bone marrow involvement * NOTE: Growth factor is not permitted within 14 days of ANC assessment unless cytopenia is secondary to disease involvement * Platelets \>= 75,000/mm\^3 unless due to bone marrow involvement, and independent of transfusion support, with no active bleeding * Direct bilirubin =\< 1.5 X upper limit of normal (ULN) (unless has Gilbert's disease or compensated hemolysis directly attributable to CLL) * Aspartate aminotransferase (AST) =\< 2.5 X ULN * Alanine aminotransferase (ALT) =\< 2.5 X ULN * Estimated creatinine clearance of \>= 30 mL/min using the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) 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 or abstain from heterosexual activity for the course of the study through at least 3 months after the last dose of protocol therapy * A woman is considered of childbearing potential, ie, fertile, following menarche and until becoming postmenopausal unless permanently sterile. Permanent sterilization methods include hysterectomy, bilateral salpingectomy, and bilateral oophorectomy. Contraception methods include the following: * Combined (estrogen- and progestogen- containing) hormonal contraception associated with the inhibition of ovulation or oral, intravaginal, or transdermal * Progestogen-only hormonal contraception associated with the inhibition of ovulation or oral, injectable, implantable * An intrauterine device * Intrauterine hormone-releasing system * Bilateral tubal occlusion vasectomized partner (provided that the vasectomized partner is the sole sexual partner of the woman of childbearing potential study participant and that the vasectomized partner has received medical assessment of surgical success) * Sexual abstinence (defined as refraining from heterosexual intercourse during the entire period of risk associated with the study treatment, starting the day prior to first dose of study drug, for the duration of the study, and for \>= 90 days after the last dose of zanubrutinib or ibrutinib. 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 (eg, calendar, ovulation, symptothermal, postovulation 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. If patient is using hormonal contraceptives such as birth control pills or devices, a barrier method of contraception (eg, condoms) must also be used. A postmenopausal state is defined as no menses for 12 months without an alternative medical cause. A high follicle-stimulating hormone level in the postmenopausal range may be used to confirm a postmenopausal state in women not using hormonal contraception or hormonal replacement therapy. However, in the absence of 12 months of amenorrhea, a single follicle-stimulating hormone measurement is insufficient Exclusion Criteria: * Chronic use of corticosteroids in excess of 20 mg/day prednisone or its equivalent * Major surgery (under general anesthesia) within 30 days prior to therapy * Uncontrolled coagulopathy or bleeding disorder. Direct oral anticoagulants are allowed * Use of moderate or strong cytochrome P450 3A4 (CYP3A4) inducer within 2 weeks of the first day of study therapy. CYP3A inhibitors are allowed. * For patients intended to enroll on dose level (DL) -1 use of strong CYP3A inhibitors will be prohibited on-therapy and their use must be stopped at minimum 2 weeks prior to the first day of study therapy * Exposure to vaccination with live vaccine within 30 days prior to cycle 1 day 1 (C1D1), or anticipated need for such vaccination during treatment * History of prior malignancy except: * Malignancy treated with curative intent and no known active disease present for \>= 2 years prior to initiation of therapy on current study * Adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease * Adequately treated in situ carcinomas (e.g., cervical, esophageal, etc.) without evidence of disease * Asymptomatic prostate cancer managed with "watch and wait" strategy * Uncontrolled immune hemolysis or thrombocytopenia (positive direct antiglobulin test in absence of hemolysis or history of immune-mediated cytopenias are not exclusions) * Known positive test result for SARS-CoV-2 unless follow-up test was negative or investigator deems the infection is fully resolved * Known positive test result for hepatitis C (hepatitis C virus \[HCV\] antibody serology testing) and a positive test result for HCV ribonucleic acid (RNA). Participants with positive serology are eligible in case of negative HCV RNA test results * Known positive test result for chronic hepatitis B virus (HBV) infection (defined by hepatitis B virus surface antigen \[HBsAg\] positivity) * Participants with occult or prior HBV infection (defined as negative HBsAg and positive total hepatitis B core antibody) may be included if HBV deoxyribonucleic acid (DNA) was undetectable, provided that they are willing to undergo ongoing DNA testing. * Antiviral prophylaxis may be administered as per institutional guidelines. * Participants who have protective titers of HBsAb after vaccination or prior but cured hepatitis B are eligible * Known seropositive for or history of active viral infection with human immunodeficiency virus (HIV) * Clinically significant cardiovascular disease including the following: * Myocardial infarction within 6 months before screening * Unstable angina within 3 months before screening * New York Heart Association class III or IV congestive heart failure * History of clinically significant arrhythmias (eg, sustained ventricular tachycardia, ventricular fibrillation, Torsades de Pointes) * QTcF \> 480 milliseconds based on Fridericia's formula * History of Mobitz II second-degree or third-degree heart block without a permanent pacemaker in place * Known severe chronic obstructive pulmonary disease (COPD) * Known hepatic cirrhosis or severe pre-existing hepatic impairment * Major surgery (requiring general anesthesia) within 14 days before the first dose of study drug or a scheduled surgery during study period * Patients with clinically significant medical condition of malabsorption, inflammatory bowel disease, chronic conditions which manifest with diarrhea, refractory nausea, vomiting or any other condition that will interfere significantly with drug absorption * Females only: Pregnant or breastfeeding * 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)