Oblimersen and Imatinib Mesylate in Treating Patients With Chronic Myelogenous Leukemia

Inclusion Criteria: * Diagnosis of chronic myeloid leukemia (CML) in chronic phase; clonal cytogenetic evolution alone does not exclude patients * Patients in whom a Philadelphia chromosome \[t(9;22)\] or a variant translocation is not detectable by cytogenetic studies are eligible if they meet one of the following criteria: * Polymerase chain reaction (PCR) positive fusion transcripts for BCR/ABL; * BCR/ABL translocation present by fluorescence in situ hybridization (FISH) * Patients must have received prior therapy with imatinib mesylate (\>= 400 mg/day for \> 8 weeks without a complete hematologic response or \>= 400 mg/day for \> 6 months without a major cytogenetic response) and must not have evidence of progressive disease (accelerated or blast phases) * Patients must have received a stable dose of imatinib mesylate \>= 600 mg/day for at least 4 weeks without \> grade 1 toxicities; the first six patients enrolled will be restricted to receiving an imatinib mesylate dose of 600 mg/day while on study * No prior therapy with hydroxyurea, cytarabine, interferon, anagrelide, homoharringtonine, or any other investigational agent within 4 weeks of study enrollment * Patients may not have received other antineoplastic medications (e.g., busulfan) * No prior stem cell transplantation * Patients must not require oral anticoagulant therapy * Non-pregnant and non-nursing; treatment under this protocol would expose an unborn child to significant risks. Women and men of reproductive potential should agree to use an effective means of birth control throughout the duration of protocol treatment and for at least three months after the last dose of imatinib mesylate * No other serious illnesses which would limit survival to \< 2 years, or a psychiatric condition which would prevent compliance with treatment or informed consent * No uncontrolled cardiovascular disease, diabetes, pulmonary disease, or infection, which in the opinion of the treating physician, would make this protocol treatment unreasonably hazardous for the patient * Patients with a "currently active" second malignancy other than non-melanoma skin cancers are not eligible; patients are not considered to have a "currently active" malignancy if they have completed therapy and are considered by their physician to be at less than 30% risk of relapse within one year * Bilirubin =\< 2 mg/dL * Creatinine =\< 2 mg/dL * AST =\< 1.5 x Upper Limit of Normal * PTT =\< 1.5 x Upper limit of Normal * BHCG Negative (if patient of childbearing potential)