Inclusion Criteria:

* Patients must be enrolled on APEC14B1 and consented to Eligibility Screening on the Part A consent form prior to enrollment on AALL1131
* White Blood Cell Count (WBC) Criteria

* Age 1-9.99 years: WBC \>= 50 000/uL
* Age 10-30.99 years: Any WBC
* Age 1-30.99 years: Any WBC with:

* Testicular leukemia
* CNS leukemia (CNS3)
* Steroid pretreatment
* Patients must have newly diagnosed B lymphoblastic leukemia (2008 World Health Organization \[WHO\] classification) (also termed B-precursor acute lymphoblastic leukemia); patients with Down syndrome are also eligible
* Organ function requirements for patients with Ph-like ALL and a predicted TKI-sensitive mutation: patients identified as Ph-like with a TKI-sensitive kinase mutation must have assessment of organ function performed within 3 days of study entry onto the dasatinib arm of AALL1131
* Creatinine clearance or radioisotope glomerular filtration rate (GFR) \> 70mL/min/1.73 m\^2 or a serum creatinine based on age/gender as follows:

* Age: Maximum Serum Creatinine (mg/dL)
* 1 to \< 6 months: 0.4 (male) 0.4 (female)
* 6 months to \< 1 year: 0.5 (male) 0.5 (female)
* 1 to \< 2 years: 0.6 (male) 0.6 (female)
* 2 \< 6 years: 0.8 (male) 0.8 (female)
* 6 to \< 10 years: 1.0 (male) 1.0 (female)
* 10 to \< 13 years: 1.2 (male) 1.2 (female)
* 13 to \< 16 years: 1.5 (male) 1.4 (female)
* \> 16 years: 1.7 (male) 1.4 (female)
* Direct bilirubin =\< 3 x upper limit of normal (ULN) for age, and
* Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 10 x upper limit of normal (ULN) for age
* Shortening fraction \>= 27% by echocardiogram, or ejection fraction \>= 50% by gated radionuclide study

* Patients must have an electrocardiogram (EKG) fewer than 6 days prior to enrollment on the dasatinib arm; patients who have had cardiac assessments by echocardiogram or radionuclide scan at the beginning of induction do not need to have these repeated prior to study entry; correct QT interval (QTc) \< 450 msec on baseline electrocardiogram as measured by the Friderica or Bazett formula
* No major conduction abnormality (unless a cardiac pacemaker is present)
* No evidence of dyspnea at rest, no exercise intolerance, and a pulse oximetry \> 94% at sea level if there is clinical indication for determination
* Patients with seizure disorder may be enrolled if on anticonvulsants and well controlled; however, drugs that induce CYP3A4/5 (carbamazepine, oxcarbazepine, phenytoin, primidone, phenobarbital) should be avoided
* Eligibility criteria for the Longitudinal, Computerized Assessment of Neurocognitive Functioning study

* Patients must be aged 6 to 13 years at time of B-ALL diagnosis, enrolled on AALL1131
* Patients must be English-, French- or Spanish-speaking (languages in which the assessment is available)
* Patients must have no known history of neurodevelopmental disorder prior to diagnosis of B-ALL (e.g., Down syndrome, Fragile X, William's Syndrome, mental retardation)
* Patients must have no significant visual impairment that would prevent computer use and recognition of the visual test stimuli
* Eligibility criteria for the National Cancer Institute (NCI) standard risk patients from AALL0932 enrolling on this study at the end of Induction
* Effective March 19, 2018, patients enrolled on AALL0932, without Down syndrome, meeting the following criteria will NOT be eligible to continue on AALL0932 or the HR B-ALL stratum of this study at the end of Induction:

* Without favorable cytogenetics (no ETV6-RUNX1 or double trisomies 4+10), with day 8 peripheral blood (PB) minimal residual disease (MRD) \>= 1% and day 29 bone marrow (BM) MRD \< 0.01%
* With favorable cytogenetics (ETV6-RUNX1 or double trisomies 4+10), with any day 8 PB MRD and day 29 BM MRD \>= 0.01%
* Both NCI standard risk (SR) and HR patients without Down syndrome and with testicular disease at diagnosis, who do not meet other VHR criteria
* Effective Amendment 6, patients enrolled on AALL0932, without Down syndrome, meeting the following criteria will NOT be eligible to continue on AALL0932 or the VHR stratum of AALL1131:

* Intrachromosomal amplification of chromosome 21 (iAMP21)
* Mixed-lineage leukemia (MLL) rearrangement
* Hypodiploidy (n \< 44 chromosomes and/or a deoxyribonucleic acid \[DNA\] index \< 0.81)
* Induction failure (M3 BM at day 29)
* Without favorable cytogenetics (no ETV6-RUNX1 or double trisomies 4+10), with day 29 BM MRD \>= 0.01%
* Patients enrolled on AALL0932, with Down syndrome, meeting the following criteria will NOT be eligible to continue on AALL0932 but WILL BE eligible to enroll on the DS HR B-ALL stratum of this study at the end of Induction:

* Day 29 MRD \>= 0.01%
* MLL rearrangement
* Hypodiploidy (n \< 45 chromosomes and/or DNA index \< 0.81)
* DS HR B-ALL patients initially enrolled on AALL0932 or this study who have Induction failure (M3 BM day 29) or Philadelphia chromosome (BCR-ABL1) will not be eligible for post-Induction therapy on either trial (AALL0932 or AALL1131)
* All patients and/or their parents or legal guardians must sign a written informed consent
* All institutional, Food and Drug Administration (FDA), and NCI requirements for human studies must be met

Exclusion Criteria:

* With the exception of steroid pretreatment or the administration of intrathecal cytarabine, patients must not have received any prior cytotoxic chemotherapy for either the current diagnosis of B-ALL or any cancer diagnosed prior to the initiation of protocol therapy on AALL1131; patients cannot have secondary B-ALL that developed after treatment of a prior malignancy with cytotoxic chemotherapy; patients receiving prior steroid therapy may be eligible for AALL1131
* Patients with BCR-ABL1 fusion are not eligible for post-induction therapy on this study but may be eligible to enroll in a successor Children's Oncology Group (COG) Philadelphia positive (Ph+) ALL trial by day 15 Induction
* DS HR B-ALL patients with Induction failure or BCR-ABL1
* Female patients who are pregnant are ineligible since fetal toxicities and teratogenic effects have been noted for several of the study drugs
* Lactating females are not eligible unless they have agreed not to breastfeed their infant
* Female patients of childbearing potential are not eligible unless a negative pregnancy test result has been obtained
* Sexually active patients of reproductive potential are not eligible unless they have agreed to use an effective contraceptive method for the duration of their study participation