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 written informed consent and Health Insurance Portability and Accountability Act (HIPAA) consent

Exclusion Criteria:

* Patients with prior chemotherapy for any malignant disorder, thoracic radiotherapy or prior surgical resection of an esophageal tumor are ineligible
* Patients with biopsy-proven invasion of the tracheobronchial tree or tracheo-esophageal fistula are ineligible
* Patients with a history of a curatively treated malignancy must be disease-free for at least two years and have a survival prognosis that is greater than five years
* Eligible patients of reproductive potential (both sexes) must agree to use an accepted and effective method of contraceptive during study therapy and for at least 6 months after the completion of bevacizumab; women must not be pregnant or breast-feeding because the study drugs administered may cause harm to an unborn fetus or breastfeeding child; all females of childbearing potential must have a serum pregnancy test to rule out pregnancy within 7 days prior to registration
* Patients with a history of hypertension must measure \< 150/90 mmHg and be on a stable regimen of anti-hypertensive therapy; patients with a history of hypertension who have a blood pressure of 150/90 mmHg, or greater are not eligible; patients with a history of hypertension who have a blood pressure of \< 150/90 mmHg but are not on a stable regimen of anti-hypertensive therapy, are not eligible
* Any prior history of hypertensive crisis or hypertensive encephalopathy
* New York Association (NYHA) grade II or greater congestive heart failure
* Patients must not have a serious or non-healing wound, skin ulcers or unhealed bone fracture, or known human immunodeficiency virus (HIV) infection
* Patients with \>= grade 2 neuropathy are not eligible
* Patients must not have had significant traumatic injury within 28 days prior to randomization
* Patients with PT (INR) \> 1.5 are not eligible; the patient may not be receiving full-dose anticoagulation; prophylactic or full dose anticoagulation are permitted post-resection or for treatment of an intercurrent thrombotic event
* Patients with non-malignant systemic disease (cardiovascular, renal, hepatic, etc.) that would preclude any of the study therapy drugs are not eligible; specifically excluded are the following conditions: current symptomatic arrhythmia, symptomatic peripheral vascular disease
* Patients with a history of the following within 12 months of study entry are not eligible: arterial thromboembolic events, unstable angina
* Any history of stroke or transient ischemic attack
* Significant vascular disease (i.e. aortic dissection, aortic aneurysm)
* Patients with psychiatric or addictive disorders or other conditions that, in the opinion of the investigator, would preclude them from meeting the study requirements are not eligible
* Distant metastases
* History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 6 months prior to study enrollment
* Known hypersensitivity to any component of bevacizumab