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 (Kar…