Inclusion Criteria:  Diagnosis of indolent B-cell non-Hodgkin lymphoma with no prior systemic therapy.  * Eligible histologies based on 2016 World Health Organization (WHO) classification include:  Follicular lymphoma (grade 1-2 or 3a) Marginal zone lymphoma. Patients with mucosa-associated lymphoid tissue (MALT) subtype of marginal zone lymphoma (MZL) may have relapsed or refractory disease after a course of antibiotic therapy  Meet criteria for initiation of therapy that include one of the following:  Symptomatic disease (including but not limited to pain/discomfort, b-symptoms) Threatened end-organ function Progressive cytopenias (leukopenia [WBC < 1,000/uL] OR hemoglobin < 10 g/dL OR platelets < 100,000/uL) Steady progression Bulky disease (one site at least 7 cm or at least four sites of 3 cm) Hepatomegaly Splenomegaly Be willing and able to provide written informed consent for the trial Have had an informed discussion with the investigator as part of the consenting/screening process that included information on treatments for these conditions with known clinical benefit, and there is documented understanding that the patient is forgoing approved available therapies Be >= 18 years of age on day of signing informed consent Have measurable fludeoxyglucose F-18 (FDG)-avid nodal disease, including at least 1 disease site measuring at least 1.5 cm in longest dimension on computed tomography (CT) or FDG-positron emission tomography (PET), or FDG-avid extra nodal measurable site measuring at least 1.0 cm in longest dimension Have a performance status of 0-2 on the ECOG Performance Scale (PS) Absolute neutrophil count (ANC) >= 1,000/uL except in cases of marrow infiltration by lymphoma Platelets >= 75,000/mcL except in cases of marrow infiltration by lymphoma or hypersplenism Hemoglobin >= 8 g/dL except in cases of marrow infiltration by lymphoma without red blood cell (RBC) transfusion within 14 days of first treatment Measured or calculated creatinine clearance (glomerular filtration rate [GFR] can also be used in place of creatinine or creatinine clearance [CrCl]) >= 40 mL/min; Note: Creatinine clearance should be calculated per institutional standard Serum total bilirubin =< 1.5 X upper limit of normal (ULN) (Patients with documented Gilbert disease may be enrolled if total bilirubin =< 3.0 x ULN) or OR direct bilirubin =< ULN for subjects with total bilirubin levels > 1.5 ULN Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT]) and alanine aminotransferase (ALT)] (serum glutamate pyruvate transaminase [SGPT]) =< 2.5 X ULN OR =< 5 X ULN for subjects with liver involvement International normalized ratio (INR) or prothrombin time (PT) =< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants Activated partial thromboplastin time (aPTT) =< 1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants, or subject is shown to have an antiphospholipid antibody on workup For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods that result in a failure rate of =< 1% per year during the treatment period and for at least 3 months after the last dose mosunutuzimab or 6 months after the last dose of obinutuzumab. Women must refrain from donating eggs during this same period. A woman is considered to be of childbearing potential if she is post-menarcheal, has not reached a postmenopausal state ( =< 12 continuous months of amenorrhea with no identified cause other than menopause), and has not undergone surgical sterilization (removal of ovaries and/or uterus). The definition of childbearing potential may be adapted for alignment with local guidelines or requirements. Examples of contraceptive methods with a failure rate of =< 1% per year include bilateral tubal ligation, male sterilization, hormonal contraceptives that inhibit ovulation, hormone-releasing intrauterine devices, and copper intrauterine devices. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post-ovulation methods) and withdrawal are not acceptable methods of contraception For women of childbearing potential, a negative serum pregnancy test result during screening period. Women who are considered not to be of childbearing potential are not required to have a pregnancy test For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use a condom, and agreement to refrain from donating sperm, as defined below: With female partners of childbearing potential or pregnant female partners, men must remain abstinent or use a condom during the treatment period and for at least 5 months after the last treatment. Men must refrain from donating sperm during this same period. The reliability of sexual abstinence should be evaluated in relation to the duration of the clinical trial and the preferred and usual lifestyle of the patient. Periodic abstinence (e.g., calendar, ovulation, symptothermal, or post ovulation methods) and withdrawal are not acceptable methods of preventing drug exposure. Male patients considering preservation of fertility should bank sperm before study treatment Patients on agents that modulate CYP3A4 should be aware that these agents are prohibited if they require treatment in Part B, and require discontinuation for at least 5 half-lives in order to proceed with Part B  Exclusion Criteria:  Contraindication to any of the individual components of this regimen or history of severe allergic or anaphylactic reactions to humanized or murine monoclonal antibodies, or known sensitivity or allergy to murine products Prior systemic treatment for lymphoma with the exception of corticosteroids as outlined below. Prior radiotherapy is allowed provided that this site is not used as a measurable site to assess response Absolute lymphocyte count > 5000/uL  History of autoimmune disease, including but not limited to myocarditis, pneumonitis, myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Guillain Barre syndrome, multiple sclerosis, vasculitis, or glomerulonephritis  Patients with a remote history of, or well-controlled autoimmune disease, may be eligible to enroll after discussion with and confirmation by the principal investigator. Patients with controlled Type 1 diabetes mellitus who are on an insulin regimen are eligible for the study Patients with a history of autoimmune-related hypothyroidism on a stable dose of thyroid replacement hormone may be eligible for this study Patients with a history of disease-related immune thrombocytopenic purpura or autoimmune hemolytic anemia may be eligible for this study  Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible for the study provided all of following conditions are met:  Rash must cover < 10% of body surface area. Disease is well controlled at baseline and requires only low-potency topical corticosteroids. No occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high potency or oral corticosteroids within the previous 12 months Prior solid organ transplantation Current grade >1 peripheral neuropathy by clinical examination or demyelinating form of Charcot-Marie-Tooth disease Prior use of any monoclonal antibody within 3 months of the start of cycle 1; any investigational therapy within 28 days prior to the start of cycle 1; vaccination with live vaccines within 28 days prior the start of cycle 1 Prior corticosteroid use for conditions related or unrelated to lymphoma are allowed provided that at least 14 days have lapsed since last dose and initiation of study therapy  History of other malignancy that could affect compliance with the protocol or interpretation of results except with permission of the principal investigator. The following are eligible without a specific waiver:  Patients with a history of curatively treated basal or squamous cell carcinoma or melanoma of the skin or in situ carcinoma of the cervix at any time prior to the study are eligible. Patients with any malignancy appropriately treated with curative intent and the malignancy has been in remission without treatment for >= 2 years prior to enrollment are 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 Evidence of significant, uncontrolled, concomitant diseases 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 last 6 months, unstable arrhythmias, or unstable angina) or pulmonary disease (including obstructive pulmonary disease and history of bronchospasm) Recent major surgery (within 4 weeks prior to the start of cycle 1), other than for diagnosis History or presence of an abnormal electrocardiogram (ECG) that is clinically significant in the investigator's opinion Known active bacterial, viral, fungal, mycobacterial, parasitic, or other infection (excluding fungal infections of nail beds) which requires systemic treatment. Patients may proceed with screening during treatment for infection, but systemic treatment must be completed by cycle 1 day 1  Positive test results for chronic hepatitis B infection (defined as positive hepatitis B surface antigen [HBsAg] serology):  * Patients with occult or prior hepatitis B infection (defined as positive total hepatitis B core antibody and negative HBsAg) may be included if hepatitis B virus (HBV) deoxyribonucleic acid (DNA) is undetectable at the time of screening. These patients must be willing to undergo monthly DNA testing and appropriate antiviral therapy as indicated by institutional standard  Positive test results for hepatitis C (hepatitis C virus (HCV) antibody serology testing)  * Patients positive for HCV antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA)  History of uncontrolled human immunodeficiency virus (HIV)  * Patients with known diagnosis of HIV must have undetectable viral load and be on antiretroviral therapy  Patients with a history of progressive multifocal leukoencephalopathy History of known central nervous system involvement History of chronic active Epstein-Barr Virus (EBV) History of hemophagocytic lymphohistiocytosis (HLH) or macrophage activation syndrome (MAS) Pregnancy or lactation or intending to become pregnant during study