Imatinib Mesylate or Dasatinib in Treating Patients With Previously Untreated Chronic Phase Chronic Myelogenous Leukemia

Inclusion Criteria: * Patients must have CML in chronic phase based on bone marrow aspiration and biopsy and peripheral blood counts obtained within 28 days before registration * Patients must be registered on this study within 180 days after the date of first being diagnosed with CML, based on a cytogenetic or molecular analysis of peripheral blood or bone marrow, demonstrating the presence of the Philadelphia chromosome or variants of the (9;22) translocation or testing positive for Bcr-Abl by real time-polymerase chain reaction (RT-PCR); patients may have secondary chromosomal abnormalities in addition to the Philadelphia chromosome and remain eligible * Patients must have Zubrod performance status 0-2 * Patients must not have received prior treatment for CML with the exception of hydroxyurea and/or anagrelide * Patients must not have received any prior chemotherapy regimen for peripheral blood stem cell mobilization; (prior collection of unmobilized peripheral blood stem cells is permitted) * Serum bilirubin =\< 2.0 x the institutional upper limit of the normal (IULN) * Serum glutamic oxaloacetic transaminase (SGOT)/aspartate aminotransferase (AST) or serum glutamate pyruvate transaminase (SGPT)/alanine transaminase (ALT) =\< 2.0 x the IULN * Patients (Southwestern Oncology Group \[SWOG\] institutions only) must be registered on SWOG-9007, "Cytogenetic Studies in Leukemia Patients;" collection of the pretreatment bone marrow specimen must be completed within 28 days prio…