Inclusion Criteria:  Diagnosis of SLE according to the 2019 European League Against Rheumatism/American College of Rheumatology Classification Criteria at least 12 weeks or more prior to screening Presence of one or more of the following SLE autoantibodies documented within the 12 months prior to screening or during screening: positive ANA (greater than or equal to 1:160); anti dsDNA above the upper limit of normal (ULN); anti-Sm above the ULN Active SLE disease, as demonstrated by a SLEDAI-2K total score of greater than or equal to 4 at screening Current receipt of one or more of the following classes of standard therapies for the treatment of SLE at stable doses: oral corticosteroids (OCSs), antimalarial agents, conventional immunosuppressants For women of childbearing potential: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraception as defined by the protocol For men on mycophenolate mofetil (MMF): With a female partner of childbearing potential, men who are not surgically sterile must remain abstinent (refrain from heterosexual intercourse) or use contraception as defined by the protocol  Exclusion Criteria:  Pregnant or breastfeeding, or intending to become pregnant during the study or within 3 months after the final dose of mosunetuzumab and 3 months after the final dose of tocilizumab Active severe or unstable lupus-associated neuropsychiatric disease that is likely to require treatment with protocol-prohibited therapies Active overlap syndrome with mixed connective tissue disease or systemic sclerosis within 12 months prior to screening or during screening Catastrophic or severe antiphospholipid syndrome within 12 months prior to screening or during screening Presence of significant lupus-associated renal disease and/or renal impairment that is likely to require treatment with protocol-prohibited therapies Peripheral CD19+ B-cell count < 25 cells/uL Receipt of an investigational therapy within 30 days or 5 drug-elimination half-lives (whichever is longer) prior to initiation of study treatment and during the study Receipt of any of the following excluded therapies: any anti-CD19 or anti-CD20 therapy such as blinatumomab, obinutuzumab, rituxumab, ocrelizumab, or ofatumumab less than 12 months prior to screening or during screening; inhibitors of JAK, Bruton tyrosine kinase, or tyrosine kinase 2, including baricitinib, tofacitinib, upadacitinib, filgotinib, ibrutinib, or fenebrutinib, or any investigational agent within 30 days prior to screening or during screening; tacrolimus, ciclosporin, or voclosporin within 30 day prior to screening or during screening; cyclophosphamide or a biologic therapy such as but not limited to belimumab, ustekinumab, anifrolumab, secukinumab, or atacicept during 2 months prior to screening or during screening; any live or attenuated vaccine during 28 days prior to screening or during screening High risk for any clinically significant bleeding or any condition requiring plasmapheresis, IV immunoglobulin, or acute blood product transfusions Significant or uncontrolled medical disease that would preclude participation HIV infection, acute or chronic hepatitis B virus (HBV), acute or chronic hepatitis C (HCV) infection, tuberculosis (TB) infection, known or suspected chronic active Epstein-Barr virus (EBV) infection, or cytomegalovirus (CMV) infection Active infection of any kind, excluding fungal infection of the nail beds Any major episode of infection that fulfills any of the following criteria: requires hospitalization during 8 weeks prior to screening or during screening; requires treatment with IV antibiotics (or anti-infective medications) during 8 weeks prior to screening or during screening; requires treatment with oral antibiotics (or anti-infective medications) during 2 weeks prior to screening or during screening History of serious recurrent or chronic infection History of progressive multifocal leukoencephalopathy (PML) History of cancer, including solid tumors, hematological malignancies, and carcinoma in situ, within the past 5 years Major surgery requiring hospitalization during 4 weeks prior to screening or during screening or any planned surgery or procedure requiring hospitalization during 12 weeks following study drug administration Current alcohol or drug abuse or history of alcohol or drug abuse within 12 months prior to screening or during screening Intolerance or contraindication to study therapies including history of severe allergic or anaphylactic reactions to monoclonal antibodies or known hypersensitivity to any component of mosunetuzumab injection Positive serum human chorionic gonadotropin measures at screening Any of the following laboratory parameters: aspartate transaminase (AST) or alanine transaminase (ALT) > 2.5 x upper limit of normal (ULN); total bilirubin > 1.5 x ULN; absolute neutrophil count (ANC) < 2.0 x 10^9/L (< 2000/mm^3); platelet count < 100 x 10^9/L (100,000 mm^3); hemoglobin < 100 g/L (10 g/dL); estimated glomerular filtration rate (eGFR) < 30 ml/min/1.73 m^2 calculated according to the Chronic Kidney Disease Epidemiology Collaboration equation; positive serum human chorionic gonadotropin measured at screening