Bevacizumab and Combination Chemotherapy Before Surgery in Treating Patients With Locally Advanced Esophageal or Stomach Cancer

Inclusion Criteria: * Patients must have biopsy proven adenocarcinoma, squamous cell carcinoma or undifferentiated carcinoma of the esophagus, GE junction and/or gastric cardia * Patients must have potentially resectable disease by the thoracic, minimally invasive or transhiatal approach * No portion of the lesion may be within 5 cm of the cricopharyngeus * Patient must be considered medically fit for surgery with average or below average risk * T1-3 or T4 with local invasion confined to diaphragm, pleura or pericardium * No myocardial infarction within 12 months of enrollment * Patients must have Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * White blood cells (WBC) \>= 3,500/mm\^3 * Platelet count \>= 100,000/mm\^3 * Serum creatinine (Cr) =\< 1.5 mg and/or creatinine clearance \>= 60 cc/min * Bilirubin must be \< upper limit of normal (ULN) unless the patient has a chronic grade 1 bilirubin elevation due to Gilbert's disease or similar syndrome due to slow conjugation of bilirubin * Alkaline phosphatase must be \< ULN * Aspartate aminotransferase (AST) \& alanine aminotransferase (ALT) must be \< ULN * Urine protein/creatinine (UPC) ratio of \< 1.0 or dipstick for protein of \< 2+, Common Terminology Criteria for Adverse Events version 4.0 (CTCAE v 4) grade \< 2; patients with a UPC ratio \>= 1.0 or dipstick of 2+ must undergo a 24-hour urine collection and must demonstrate \< 1 gm of protein in order to participate * Patients must give writ…