Inclusion Criteria:

* INDUCTION/CONSOLIDATION REGISTRATION:
* Patients must have a morphologic diagnosis of acute lymphoblastic leukemia (ALL), with evidence of ALL involvement in bone marrow and/or blood; patients with only extramedullary disease in the absence of bone marrow or blood involvement are not eligible; patients with M0 acute myeloid leukemia (AML) or mixed lineage leukemia are not eligible for this study; patients with L3 (Burkitts) are also not eligible

* For ALL in marrow or peripheral blood, immunophenotyping of the blood or marrow lymphoblasts must be performed to determine lineage (B cell, T-cell, or mixed B/T cell); NOTE: appropriate marker studies including cluster of differentiation (CD)19 (B cell), CD10, CD5, and CD7 (T cell) must be performed; co-expression of myeloid antigens (CD13 and CD33) will not exclude patients; if possible, the lineage specific markers cytoplasmic CD22 or CD79a (B cells), cytoplasmic CD3 (T cells) and cytoplasmic myeloperoxidase (MPO) (myeloid cells) must be determined
* Patients may have received no more than one course of remission induction therapy for ALL; patients who have received any post-remission therapy for ALL or who have relapsed from complete remission are not eligible; (patients with previously untreated ALL can be eligible, and patients who have received one course of remission induction therapy for ALL can be eligible, regardless of their response to therapy); patients may have received no more than 14 days of tyrosine kinase inhibitor therapy prior to registration; any prior induction chemotherapy must have been completed no more than 28 days prior to registration

* NOTE: If the patient has been initiated on the protocol defined regimen (i.e. the hyper-CVAD regimen without a tyrosine kinase inhibitor) before the Philadelphia chromosome (Ph)/BCR-ABL status was known, the patient may be registered on the protocol and start dasatinib; in this first course, dasatinib will be administered up to day 14 (i.e. if the patient is registered on day 5 and starts therapy on day 6, only 8 days of dasatinib will be administered and dasatinib will be completed on day 14)
* For patients who have received any prior therapy that was NOT remission induction therapy, one of the following must be true:

* At least 6 weeks must have elapsed since any monoclonal antibodies were given, at least 7 days must have elapsed since any other treatment was given, and all toxicities of the remission induction therapy must have resolved to grade =\< 2
* The patient must have rapidly progressive disease (per institutional guidelines)
* For previously treated patients, the study chair must be contacted before registration, in order to determine the regimen to be given in the first course of induction/consolidation therapy, based on prior therapy
* Patients must be Philadelphia (Ph) positive and/or BCR/ABL positive as confirmed by standard cytogenetics, fluorescent in situ hybridization (FISH), and/or polymerase chain reaction (PCR) testing performed by local laboratory; NOTE: samples will be submitted centrally for verification of results
* Patients must have a bilirubin =\< 3.0 x institutional upper limit of normal (IULN) within 14 days prior to registration
* Patients must have serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 3.0 x IULN and/or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 3.0 x IULN within 14 days prior to registration; if both tests are done then both values must be =\< 3.0 x IULN
* Patients must have a serum creatinine =\< 3.0 x IULN within 14 days prior to registration
* Patients must not have active pericardial effusion, ascites, or pleural effusion of any grade; exception: if the effusion is suspected to be related to the leukemia, the patient may have pericardial effusion of =\< grade 2 or pleural effusion =\< grade 1
* Patients may not have any clinically significant cardiovascular disease including the following:

* Myocardial infarction or ventricular tachyarrhythmia within 6 months
* Prolonged corrected QT (QTc) \>= 480 msec (Fridericia correction)
* Ejection fraction less than institutional normal
* Major conduction abnormality (unless a cardiac pacemaker is present)
* Patients with any cardiopulmonary symptoms of unknown cause (e.g. shortness of breath, chest pain, etc.) should be evaluated by a baseline echocardiogram with or without stress test as needed in addition to electrocardiogram (EKG) to rule out QTc prolongation; the patient may be referred to a cardiologist at the discretion of the principal investigator; patients with underlying cardiopulmonary dysfunction should be excluded from the study
* Patients must have Zubrod performance status of 0-2
* 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 5 years
* Collection and submission of pre-treatment cytogenetic specimens must be completed within 28 days prior to registration on S0805
* Collection and submission of pretreatment marrow and/or peripheral blood specimens for cellular and molecular studies, including verification of BCR/ABL status must be completed within 28 days prior to registration
* Patients must not be pregnant or nursing; women/men of reproductive potential must have agreed to use an effective contraceptive method; a woman is considered to be of "reproductive potential" if she has had menses at any time in the preceding 12 consecutive months; in addition to routine contraceptive methods, "effective contraception" also includes heterosexual celibacy and surgery intended to prevent pregnancy (or with a side-effect of pregnancy prevention) defined as a hysterectomy, bilateral oophorectomy or bilateral tubal ligation; however, if at any point a previously celibate patient chooses to become heterosexually active during the time period for use of contraceptive measures outlined in the protocol, he/she is responsible for beginning contraceptive measures
* Patients must not have prior history of known type I hypersensitivity or anaphylactic reactions to doxorubicin
* Patients or their legally authorized representative 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
* MAINTENANCE/INTENSIFICATION:
* Patient must have achieved CR or CRi within 2 courses of Induction/Consolidation Chemotherapy; patient must remain in CR or CRi until beginning Maintenance Chemotherapy and this must be re-documented by bone marrow and peripheral blood examination within 28 days prior to registration to Step 2
* All treatment related toxicities must have resolved to =\< grade 2
* Patients must not have received allogeneic stem cell transplant
* TRANSPLANT REGISTRATION:
* Patients must have an available completely matched sibling donor or a 10/10 matched non-sibling donor
* Patients must have allogeneic stem cell transplant arranged prior to registration to Step 3
* Patients must have documented CR or CRi within 14 days prior to registration to Step 3
* Patients must not be HIV + (human immunodeficiency virus); a negative HIV test must be obtained within 14 days prior to registration
* POST TRANSPLANT/POST-MAINTENANCE SINGLE-AGENT DASATINIB THERAPY:
* Patients must have reached day 100 post transplant or must have completed protocol maintenance/intensification (or must have been approved by the Study Chair after early removal from maintenance/intensification)
* Patients must be in CR or CRi based on bone marrow and peripheral blood examination within 28 days prior to registration to Step 4
* Patients must have recovered to =\< grade 2 from all treatment related toxicity