Fludarabine Phosphate, Rituximab, and Bevacizumab in Treating Patients With B-Cell Chronic Lymphocytic Leukemia That Has Relapsed or Not Responded To Treatment

Inclusion * Relapse or refractory chronic Lymphocytic leukemia as defined by the WHO criteria and exhibit active disease requiring treatment as per the NCI working group in CLL * Disease measurable defined by a combination of lymphocytosis \>= 5,000/mm\^3 in peripheral blood and lymphocytosis \>= 30% in bone marrow * Confirmed CD20 expression on malignant CLL cells * ECOG performance status of 0-2 * Life expectancy of at least 6 months * Documented negative serologic testing for human immunodeficiency virus (HIV), hepatitis B (unless serologically positive due to prior vaccination), and hepatitis C within the year prior to enrollment * Aspartate aminotransferase (AST) \< 2.5 x upper limit of normal (ULN) * Total serum bilirubin \< 2.5 x ULN * Serum creatinine \< 1.5 x ULN * Hemoglobin \> 8 g/dL * Absolute neutrophil count (ANC) \> 1,000 cells/mm\^3 * Platelet count \> 50,000/mm\^3 * PT/INR and PTT \< 1.5 x ULN * Within 2 weeks prior to registration, patients must have had a urinalysis negative for protein or a 24-hour urine collection demonstrating \< 500 mg protein * If female and of child-bearing potential, have a negative serum pregnancy test within 14 days of enrollment * If sexually active male or sexually active female of reproductive capability, has agreed to use a medically accepted form of contraception from time of enrollment to completion of all follow-up study visits * Signed an institutional review board (IRB)-approved informed consent document for this protocol Exclusion * Patients must not require sustained support of hematopoietic cytokines or transfusion of blood products * Presence of acute infection or other significant systemic illness * Central nervous system involvement by malignancy, history of CVA, or seizure * Previously received Bevacizumab * Received transplant or Alemtuzumab within 3 months of enrollment * Received an investigational agent, systemic corticosteroids, chemotherapy, immunotherapy, biologic therapy, antibody therapy (e.g., Rituximab) and/or radiation therapy within one month of enrollment * Radiation to \> 25% of bone marrow or any radiation therapy within 4 weeks prior to start of therapy * Inability to comply with study and/or follow-up procedures * Life expectancy of less than 6 months * Fludarabine-refractory disease (no response of disease to \>= 3 cycles of a fludarabine-based regimen or relapse within 6 months of fludarabine-based regimen) * Current, recent (within 4 weeks of the first infusion of this study), or planned participation in an experimental drug study other than a Genentech-sponsored bevacizumab cancer study * Patients with prior malignancy other than lymphoma, except for adequately-treated skin cancer (basal cell or squamous cell carcinoma), in situ cervical cancer, or other cancer for which the patient has been disease-free for 5 years unless approved by the PI * Inadequately controlled hypertension (defined as systolic blood pressure \> 150 mmHg and/or diastolic blood pressure \> 100 mmHg) * Prior history of hypertensive crisis or hypertensive encephalopathy * New York Heart Association (NYHA) Grade II or greater congestive heart failure * History of myocardial infarction or unstable angina within 6 months prior to Day 1 * History of stroke or transient ischemic attack within 6 months prior to Day 1 * Known CNS disease, except for treated brain metastasis; Patients with CNS metastases treated by neurosurgical resection or brain biopsy performed within 3 months prior to Day 1 will be excluded * Significant vascular disease (e.g., aortic aneurysm, requiring surgical repair or recent peripheral arterial thrombosis) within 6 months prior to Day 1 * History of hemoptysis (\>= 1/2 teaspoon of bright red blood per episode) within 1 month prior to Day 1 * Evidence of bleeding diathesis or significant coagulopathy (in the absence of therapeutic anticoagulation) * Major surgical procedure, open biopsy, or significant traumatic injury within 28 days prior to Day 1, or anticipation of need for major surgical procedure during the course of the study * Core biopsy or minor surgical procedure, excluding placement of a vascular access device, within 7 days prior to Day 1 * History of abdominal fistula or gastrointestinal perforation within 6 months prior to Day 1 * Serious, non-healing wound, active ulcer, or untreated bone fracture * Proteinuria as demonstrated by a UPC ratio \>= 1.0 at screening * Known hypersensitivity to any component of bevacizumab * Pregnancy (positive pregnancy test) or lactation; use of effective means of contraception (men and women) in subjects of childbearing potential