Atezolizumab and Bevacizumab in Treating Patients With Recurrent, Persistent, or Metastatic Cervical Cancer

Inclusion Criteria: * Patients must have measurable disease per RECIST 1.1; measurable lesions are defined as those that can be accurately measured in at least one dimension (longest diameter to be recorded as \>= 10 mm (\>= 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam; to be considered pathologically enlarged and measurable, a lymph node must be \>= 15 mm (\>= 1.5 cm) in short axis * Patients must have had one prior systemic chemotherapeutic regimen for management of recurrent, persistent or metastatic carcinoma of the cervix (e.g.; paclitaxel/cisplatin, paclitaxel/cisplatin/bevacizumab), at least one which must have contained bevacizumab * NOTE: Patients are allowed to receive 1-2 prior regimens for management of recurrent, persistent or metastatic carcinoma of the cervix; patients who have received more than two prior systemic regimens for management of recurrent, persistent or metastatic carcinoma of the cervix are NOT eligible * NOTE: Prior adjuvant therapy is NOT counted as a systemic chemotherapeutic regimen for management of recurrent, persistent or metastatic carcinoma of the cervix; adjuvant therapy includes cisplatin given concurrent with primary radiation therapy (CCRT) and adjuvant chemotherapy given following the completion of concurrent chemotherapy and radiation therapy (e.g., paclitaxel and carboplatin for up to 4 cycles) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (however, patients with known Gilbert disease who have serum bilirubin level =\< 3 x ULN may be enrolled) * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x ULN * Alkaline phosphatase =\< 2.5 x ULN * Creatinine =\< 1.5 x ULN * International normalized ratio (INR) and activated partial thromboplastin time aPTT =\< 1.5 x ULN (This applies only to patients who do not receive therapeutic anticoagulation; patients receiving therapeutic anticoagulation, such as low-molecular-weight heparin or warfarin, should be on a stable dose) * Urine protein must be screened by urinalysis; if protein is 2+ or higher, 24-hour urine protein should be obtained and the level should be \< 1000 mg for patient enrollment * Patient must have recurrent, persistent or metastatic cervical cancer including squamous cell, adenocarcinoma and adenosquamous histologies; mesonephric carcinoma, minimal deviation/adenoma malignum, clear cell carcinoma and gastric type are excluded * Women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 5 months (150 days) after the last dose of atezolizumab; fertile women must agree to use adequate contraceptive measures during study therapy and for at least 6 months after the completion of bevacizumab therapy; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document * Tumors within previous radiated field will be designated "non-target" lesions unless progression is documented or a biopsy is obtained to confirm persistence at least 90 days following completion of radiation therapy * Willingness to undergo a tumor biopsy Exclusion Criteria: * Patients with prior allogeneic bone marrow transplantation or prior solid organ transplantation * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events (other than alopecia) due to agents administered more than 4 weeks earlier; however, the following therapies are allowed: * Hormone-replacement therapy or oral contraceptives * Herbal therapy \> 1 week prior to cycle 1, day 1 (herbal therapy intended as anticancer therapy must be discontinued at least 1 week prior to cycle 1, day 1) * Prior treatment with anti-PD-1, or anti-PD-L1 therapeutic antibody or pathway-targeting agents * Prior treatment with anti-CTLA-4 therapeutic antibody or pathway-targeting agents * Treatment with any other investigational agent within 4 weeks prior to cycle 1, day 1 * Treatment with systemic immunostimulatory agents (including, but not limited to, interferon \[IFN\]-alpha or interleukin \[IL\]-2) within 6 weeks prior to cycle 1, day 1 * Treatment with systemic immunosuppressive medications (including, but not limited to, prednisone, cyclophosphamide, azathioprine, methotrexate, thalidomide, and anti-tumor necrosis factor \[anti-TNF\] agents) within 2 weeks prior to cycle 1, day 1 * Patients who have received acute, low dose, systemic immunosuppressant medications (e.g., a one-time dose of dexamethasone for nausea) may be enrolled * The use of inhaled corticosteroids and mineralocorticoids (e.g., fludrocortisone) for patients with orthostatic hypotension or adrenocortical insufficiency is allowed * Patients taking bisphosphonate therapy for symptomatic hypercalcemia; use of bisphosphonate therapy for other reasons (e.g., bone metastasis or osteoporosis) is allowed * Patients with known brain metastases should be excluded from this clinical trial * Known hypersensitivity to Chinese hamster ovary cell products or other recombinant human antibodies * History of severe allergic, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies or fusion proteins * History of allergic reactions attributed to compounds of similar chemical or biologic composition to bevacizumab or atezolizumab * Known clinically significant liver disease, including active viral, alcoholic, or other hepatitis; cirrhosis; fatty liver; and inherited liver disease * Patients with past or resolved hepatitis B infection (defined as having a negative hepatitis B surface antigen \[HBsAg\] test and a positive anti-HBc \[antibody to hepatitis B core antigen\] antibody test) are eligible * Patients positive for hepatitis C virus (HCV) antibody are eligible only if polymerase chain reaction (PCR) is negative for HCV ribonucleic acid (RNA) * History or risk of autoimmune disease, including, but not limited to, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, vascular thrombosis associated with antiphospholipid syndrome, Wegener's granulomatosis, Sjogren's syndrome, Bell's palsy, Guillain-Barre syndrome, multiple sclerosis, autoimmune thyroid disease, vasculitis, or glomerulonephritis * Patients with a history of autoimmune hypothyroidism on a stable dose of thyroid replacement hormone may be eligible * Patients with controlled type 1 diabetes mellitus on a stable insulin regimen may be eligible * Patients with eczema, psoriasis, lichen simplex chronicus of vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis would be excluded) are permitted provided that they meet the following conditions: * Patients with psoriasis must have a baseline ophthalmologic exam to rule out ocular manifestations * Rash must cover less than 10% of body surface area (BSA) * Disease is well controlled at baseline and only requiring low potency topical steroids (e.g., hydrocortisone 2.5%, hydrocortisone butyrate 0.1%, fluocinolone 0.01%, desonide 0.05%, alclometasone dipropionate 0.05%) * No acute exacerbations of underlying condition within the last 12 months (not requiring psoralen plus ultraviolet A radiation \[PUVA\], methotrexate, retinoids, biologic agents, oral calcineurin inhibitors; high potency or oral steroids) * History of idiopathic pulmonary fibrosis, pneumonitis (including drug induced), organizing pneumonia (i.e., bronchiolitis obliterans, cryptogenic organizing pneumonia, etc.), or evidence of active pneumonitis on screening chest computed tomography (CT) scan; history of radiation pneumonitis in the radiation field (fibrosis) is permitted * Patients with active tuberculosis (TB) are excluded * Severe infections within 4 weeks prior to cycle 1, day 1, including, but not limited to, hospitalization for complications of infection, bacteremia, or severe pneumonia * Signs or symptoms of infection within 2 weeks prior to cycle 1, day 1 * Received oral or intravenous (IV) antibiotics within 2 weeks prior to cycle 1, day 1; patients receiving prophylactic antibiotics (e.g., for prevention of a urinary tract infection or chronic obstructive pulmonary disease) are eligible * Major surgical procedure within 28 days prior to cycle 1, day 1 or anticipation of need for a major surgical procedure during the course of the study * Administration of a live, attenuated vaccine within 4 weeks before cycle 1, day 1 or anticipation that such a live, attenuated vaccine will be required during the study and up to 5 months after the last dose of atezolizumab * Influenza vaccination should be given during influenza season only (approximately October to March); patients must not receive live, attenuated influenza vaccine within 4 weeks prior to cycle 1, day 1 or at any time during the study * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Patients positive for human immunodeficiency virus (HIV) are NOT excluded from this study, but HIV-positive patients must have: * A stable regimen of highly active anti-retroviral therapy (HAART) * No requirement for concurrent antibiotics or antifungal agents for the prevention of opportunistic infections * A CD4 count above 250 cells/mcL and an undetectable HIV viral load on standard PCR-based tests * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with atezolizumab and/or bevacizumab * Malignancies other than the cervical cancer within 5 years prior to cycle 1, day 1, with the exception of those with a negligible risk of metastasis or death, such as adequately controlled basal cell carcinoma or squamous cell carcinoma of the skin or carcinoma in situ of the breast * Significant vascular disease (e.g., aortic aneurysm, requiring surgical repair or recent peripheral arterial thrombosis) within 6 months prior to day 1 * Patients with clinically significant cardiovascular disease are excluded * Inadequately controlled hypertension (HTN) (systolic blood pressure \[SBP\] \>= 160 mmHg and/or diastolic blood pressure \[DBP\] \>= 90 mmHg despite antihypertensive medication) * History of cerebrovascular accident (CVA) within 6 months * Myocardial infarction or unstable angina within 6 months * New York Heart Association class II or greater congestive heart failure * Serious and inadequately controlled cardiac arrhythmia * Significant vascular disease (e.g. aortic aneurysm, history of aortic dissection) * Clinically significant peripheral vascular disease * History of abdominal/pelvic fistula, gastrointestinal perforation and/or intraabdominal abscess within 6 months prior to day 1 * Evidence of bleeding diathesis or clinically significant coagulopathy * Serious or non-healing wound, active ulcer or bone fracture * Patients requiring treatment with a RANKL inhibitor (e.g. denosumab) who cannot discontinue it before treatment with atezolizumab