Lapatinib Ditosylate, Trastuzumab, Paclitaxel, and Surgery in Treating Patients With Breast Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed primary invasive breast cancer * Primary tumor is larger than 2 cm in diameter (T2) as measured by caliper or ultrasound * Overexpression and/or amplification of HER2 is confirmed by immunohistochemistry (IHC) 3+ or fluorescence in situ hybridization (FISH) + when IHC 2+ * Patients have not received prior therapies for breast cancer * Patients have Karnofsky \>= 70% * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Hemoglobin \>= 9.0 g/dL * Platelets \>= 75,000/mcL * Total bilirubin =\< 1.5 times institutional upper limit of normal * Aspartate aminotransferase (AST)(serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT)(serum glutamic pyruvate transaminase\[SGPT\]) =\< 2.5 times institutional ULN * Creatinine =\< 1.5 times institutional upper limit of normal (ULN) * Patients must have left ventricular ejection fraction (LVEF) \>= 50% by multi-gated acquisition (MUGA) or echocardiography * Patients must be able to take oral medications (i.e., no uncontrolled vomiting, inability to swallow, or diagnosis of chronic malabsorption) * Women of childbearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation as well as for at least 6 months after the last dose of trastuzumab * Ability to understand and willingness not only for treatment but also for undergoing serial biopsies and sign a written informed consent document * Only Japanese women are eligible for the trial Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy * Patients who are receiving any other investigational agents * Patients have distal metastasis (stage IV disease) * Patients with previous (within 10 years) or current history of malignant neoplasm except for curatively treated basal and squamous cell carcinoma of the skin or carcinoma in situ of the cervix * Patients with a history of allergic reactions attributed to compounds of similar chemical or biologic composition to lapatinib or other agents used in study * Patients receiving any medications or substances that are inhibitors or inducers of cytochrome P450 3A4 (CYP3A4) are ineligible * Patients who have uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women * Patients who have family or personal history of congenital long or short QT syndrome, Brugada syndrome, QT/QTc prolongation, or torsade de pointes * Patients who have chronic gastrointestinal disease presenting with diarrhea (inflammatory bowel disease, malabsorption, or \>= grade 2 diarrhea of any etiology at baseline) * Patients who have neuropathy \>= grade 2 of any cause * Patients are diagnosed with inflammatory breast cancer or bilateral breast cancer