Imatinib Mesylate With or Without Bevacizumab in Treating Patients With Metastatic or Unresectable Gastrointestinal Stromal Tumor

Inclusion Criteria: * REGISTRATION # 1 * Patient must have a biopsy proven diagnosis of gastrointestinal stromal tumor (GIST) that is distantly metastatic or unresectable; patients must be determined to be unresectable for cure * Patient may have measurable and/or non-measurable disease; computed tomography (CT) or magnetic resonance imaging (MRI) used for measurable disease must have been completed within 28 days prior to registration; CT or MRI used for non-measurable disease must have been completed within 42 days prior to registration; PET scans are not sufficient for disease assessment; all disease must be assessed and documented on the Baseline Tumor Assessment Form * CT/MRI scans must be performed and submitted for central review; archived tissue must be submitted as outlined * Institutions must seek additional patient consent for PET scans as outlined; if patient consents to the submission of PET scans, the patient must also be registered to Registration #2 * Patient must not have known brain metastasis * Patient must have a Zubrod performance status of 0 - 3 * Patient must have resolution of transient toxicities from any prior chemotherapy, radiation therapy or surgery to =\< grade 1 (Common Terminology Criteria for Adverse Events \[CTCAE\] version 3.0) * Patient may have previously received traditional chemotherapeutic agents in any setting, provided at least 28 days have elapsed since completing chemotherapy and they have recovered to =\< grade 1 from all drug-induced toxicities * Patient must not have received prior treatment with bevacizumab or other agents targeting VEGF, VEGFR, or PDGFR for advanced disease; those agents may have been used in the adjuvant setting if the patient did not recur for at least 12 months following the completion of treatment; patients may be receiving imatinib for advanced disease prior to registration provided they meet ALL of the following criteria: * Patient must not have received more than 30 days of imatinib treatment prior to registration * Patients have not been restaged; (baseline disease assessments prior to initiation of imatinib must fulfill requirements) * Patients must have no clinical signs of progression * Prior radiotherapy is allowed, provided at least 28 days have elapsed since the last treatment and there is evidence of progressive disease within the radiation field or disease outside the radiation field * Patient must not have had a major surgical procedure, open biopsy, or significant traumatic injury within 28 days prior to registration, or anticipation of need for major surgical procedure during the course of the study; no fine needle aspirations or core biopsies are allowed within 7 days prior to registration; no procedure to place a port-a-cath is allowed within 7 days prior to registration * Patient must have a total bilirubin =\< 2.0 x institutional upper limit of normal (IULN), obtained within 28 days prior to registration * Patients without liver involvement must have serum glutamic oxaloacetic transaminase (SGOT) or serum glutamate pyruvate transaminase (SGPT) =\< 2.5 x IULN, obtained within 28 days prior to registration; patients with liver involvement must have SGOT or SGPT =\< 5 x IULN * Patient must have adequate renal function as defined by a serum creatinine =\< 1.5 x IULN obtained within 28 days prior to registration * Patient must have urine protein/creatinine ratio (UPC) \< 1; this result must be obtained within 28 days prior to registration * Patient must have an absolute neutrophil count (ANC) \>= 1,000/mcl obtained within 28 days prior to registration * Patient must have a platelet count \>= 100,000/mcl obtained within 28 days prior to registration * Patient must have hemoglobin \>= 9 gm/dl (this may be achieved by transfusion if needed) obtained within 28 days prior to registration * Patient must have an international normalized ratio (INR) =\< 1.5, obtained within 28 days prior to registration * Patient must have a partial thromboplastin time (PTT) =\< IULN, obtained within 28 days prior to registration * Patient must not be taking therapeutic doses of Coumadin (warfarin) as anticoagulation at the time of registration; patients requiring therapeutic anticoagulation may use low-molecular weight heparin (e.g., Lovenox) or other agents, and mini-dose Coumadin (1 mg PO QD) as prophylaxis is allowed * Patient must not have had a cerebrovascular accident (CVA), transient ischemic attack (TIA), myocardial infarction or unstable angina within 6 months prior to registration; patient must not have serious cardiac arrhythmia requiring medication, New York Heart Association (NYHA) class II or greater congestive heart failure, or clinically significant peripheral vascular disease * Patient must not have had an abdominal fistula, gastrointestinal perforation or intraabdominal abscess within 28 days prior to registration * Patient must not plan to use other investigational agents while on protocol treatment * Patient must have no contraindication to oral medications (e.g., severe dysphagia); patients with gastrostomy (G)- or jejunostomy (J)- tubes are eligible * Patient must not have blood pressure \> 160/90; patients with a history of hypertension must be on a stable regimen of anti-hypertensive therapy * Patient must not have a serious, non-healing wound, ulcer, or bone fracture * Patient must not be pregnant or nursing; male and female patients of reproductive potential must agree to employ an effective barrier method of birth control throughout protocol treatment and for up to 6 months following discontinuation of study drugs * No other prior malignancy is allowed except for the following: adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, adequately treated stage I or II cancer from which the patient is currently in complete remission, or any other cancer from which the patient has been disease free for five years * If day 28 or 42 falls on a weekend or holiday, the limit may be extended to the next working day; in calculating days of tests and measurements, the day a test or measurement is done is considered day 0; therefore, if a test is done on a Monday, the Monday four weeks later would be considered day 28 * All patients must be informed of the investigational nature of this study and must sign and give written informed consent in accordance with institutional and federal guidelines * At the time of patient registration, the treating institution's name and identification (ID) number must be provided to the Data Operations Center in Seattle in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered into the data base * REGISTRATION #2 - PET SUBSTUDY: * Patient must have been registered to the main study * Patient must have consented to the submission of PET scans