Pegylated Liposomal Doxorubicin Hydrochloride and Carboplatin Followed by Surgery and Paclitaxel in Treating Patients With Triple Negative Stage II-III Breast Cancer

Inclusion Criteria: * Women with previously untreated, unilateral stage II-III breast cancer, ER/PgR/HER2 negative (ER =\< 5%, PgR =\< 5%, HER2 0-1+ by immunohistochemistry \[IHC\] or fluorescence in situ hybridization \[FISH\] =\< 2.0); if clinically negative lymph nodes, tumor size should be minimum 1.0 cm and identifiable under office-based ultrasound guidance * Negative serum or urine beta-human chorionic gonadotropin (hCG) pregnancy test at screening for patients of child-bearing potential within one week prior to enrollment * Patients with reproductive potential must use an adequate contraceptive method (e.g. abstinence, intrauterine device, oral contraceptives, barrier device with spermicide or surgical sterilization) during treatment and for three months after completing treatment * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status \< 2 * Life expectancy \>= 52 weeks excluding their diagnosis of breast cancer * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 * Platelets \>= 100,000 cells/mm\^3 * Hemoglobin \> 9.0 g/dL * Creatinine \< 2.5 mg/dL * Total bilirubin \< 1.5 X upper limit of normal (ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) \< 2.5 X institutional ULN * Patients must have normal cardiac function, as evidenced by a left ventricular ejection fraction (LVEF) within institutional normal limits; echocardiogram may be used if multi gated acquisition (MUGA) scan is not available, but the same test must be used throughout the study to evaluate LVEF * Computed tomography of the chest, abdomen, and pelvis (CT CAP) and bone scan performed within 30 days prior to study entry and does not demonstrate metastatic disease * Patients may not receive concurrent treatment with other investigational or commercial agent(s) for treatment of their breast cancer * The patient or, if applicable, her legally authorized representative must have signed and dated an Institutional Review Board (IRB)-approved consent form that conforms to federal and institutional guidelines * Patients must be eligible to undergo surgery, either lumpectomy or mastectomy for local treatment of the breast cancer; surgical margins at discretion of surgeon per National Comprehensive Cancer Network (NCCN) guidelines; axillary exploration at discretion of surgeon but all patients minimally have sentinel lymph node evaluation at time of surgery * Imaging by magnetic resonance imaging (MRI), ultrasound and/or mammogram and physical exam to document lesions size must be performed within 30 days of study entry * Must not exhibit a non-healing wound or any skin breakdown * Before administering liposomal doxorubicin, patients must wait 4-6 weeks after surgery * Submission of tumor samples from the diagnostic biopsy and breast surgery is required for all patients Exclusion Criteria: * Women who are pregnant or breastfeeding * Second primary malignancy except most situ carcinoma (e.g. in situ carcinoma of the cervix, adequately treated non-melanomatous carcinoma of the skin) or other malignancy treated at least 5 years previously with no evidence of recurrence * Definitive clinical or radiologic evidence of metastatic disease; imaging must have been performed no greater than 30 days prior to initiation of chemotherapy * Diagnosis of inflammatory breast cancer * History of hypersensitivity reactions attributed to a conventional formulation of doxorubicin hydrochloride (HCL) or the components of doxil, paclitaxel, or carboplatin * Serious concomitant systemic disorders (including active infections or chronic infection requiring suppressive antibiotics) that would compromise the safety of the patient or compromise the patient's ability to complete the study, at the discretion of the investigator * Myocardial infarct or unstable angina within 6 months before enrollment, New York Heart Association (NYHA) class II or greater heart failure, uncontrolled angina, severe uncontrolled ventricular arrhythmias, clinically significant pericardial disease, valvular disease with documented compromise in cardiac function, or electrocardiographic evidence of acute ischemic or active conduction system abnormalities * Prior anthracycline, platinum salt, or taxane for any malignancy * Known or active hepatitis B or C with abnormal liver function tests * Significant vascular disease (e.g., aortic aneurysm, aortic dissection) * Symptomatic peripheral vascular disease * Evidence of bleeding diathesis or coagulopathy * Intrinsic lung disease resulting in moderate to severe dyspnea * History of a major organ allograft or condition requiring chronic immunosuppression, e.g., kidney, liver, lung, heart, bone marrow transplant, or autoimmune diseases; this includes treatment with corticosteroids within one month (dose of \>= 10 mg/day methylprednisolone equivalent) (excluding inhaled steroids); patients who have received corneal transplants, cadaver skin, or bone transplants are eligible * Nervous system disorder (paresthesias, peripheral motor neuropathy, or peripheral sensory neuropathy) \>= grade 2, per the Common Terminology Criteria for Adverse Events version 4.0 (CTCAE v4.0) * Conditions that would prohibit administration of corticosteroids