• Patients must have histologically or cytologically confirmed alveolar soft part sarcoma that is not curable by surgery. Diagnosis of malignancy must be confirmed by the department of pathology at the institution where the patient is enrolled prior to patient enrollment. To be eligible for atezolizumab in combination with bevacizumab, the patient must have atezolizumab-refractory/resistant disease that has progressed (definitive clinical progression or confirmed progressive disease \[iCPD\]) on prior 10005 atezolizumab monotherapy

• Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) by chest x-ray or as \>= 10 mm (\>= 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam

⁃ Note: Once the primary endpoint of the trial has been met, pediatric patients may enroll with evaluable non-measurable disease and will not be required to have measurable disease. Evaluable non-measurable disease is that which is not measurable by RECIST1.1 but can be evaluated by imaging (e.g., CT, bone scan, or ultrasound) or other methods

• Patients with newly diagnosed, unresectable, metastatic and measurable ASPS will also be eligible for atezolizumab monotherapy if they show clinical evidence of disease progression (including history and increasing physical symptoms); on-study documentation will include a physician's rationale that supports evidence of clinical disease progression (i.e., increasing tumor pain)

• To be eligible for atezolizumab monotherapy, subjects must be \>= 2 years of age. To be eligible for atezolizumab in combination with bevacizumab, subjects must be \>= 18 years of age

• Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky or Lansky \>= 70%)

• Life expectancy of greater than 3 months

• Absolute neutrophil count \>= 1,500/mcL

• Platelets \>= 100,000/mcL

• Hemoglobin \>= 8 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 glutamic pyruvic transaminase \[SGPT\]) =\< 3 x ULN (AST and/or ALT =\< 5 x ULN for patients with liver involvement)

• Alkaline phosphatase =\< 2.5 x ULN (=\< 5 x ULN for patients with documented liver involvement or bone metastases)

• Creatinine clearance for adult patients (\>= 18 years of age): \>= 30 mL/min/1.73 m\^2 by Cockcroft-Gault; for pediatric patients (\< 18 years of age), a serum creatinine based on age and sex as follows:

⁃ Age 2 to \< 6 years, maximum serum creatinine (mg/dL) male 0.8, female 0.8;

• Administration of atezolizumab or bevacizumab may have an adverse effect on pregnancy and poses a risk to the human fetus, including embryo-lethality. Female patients of child-bearing potential and male patients 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 or 6 months (180 days) after the last dose of bevacizumab. Should a woman become pregnant or suspect she is pregnant while she or her partner 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 or a parent/guardian able to do the same

• Willingness to provide biopsy samples for research purposes (patients \>= 18 years of age only). At the principal investigator (PI's) discretion, archival tissue that was collected and preserved may be used in lieu of a baseline biopsy, provided no anti-cancer agents or immunotherapies were received since collection of the archival tissue

• Patients who have received live attenuated vaccines within 30 days of the first dose of trial treatment are eligible at the discretion of the investigator. All seasonal influenza vaccines and vaccines intended to prevent SARS-CoV-2 and coronavirus disease 2019 (COVID-19) are allowed

• For patients not receiving therapeutic anticoagulation: international normalized ratio (INR) or activated partial thromboplastin time (aPTT) =\< 1.5 x ULN for bevacizumab patients

• For patients receiving therapeutic anticoagulation: stable anticoagulant regimen for bevacizumab patients