Anastrozole and Letrozole After Surgery for the Treatment of Stage I-III Breast Cancer

Inclusion Criteria: * REGISTRATION-INCLUSION CRITERIA * Disease characteristics: * Histological confirmation of invasive breast carcinoma * Stage I-III breast cancer * Estrogen receptor (ER) positive disease according to American Society of Clinical Oncology (ASCO)/College of American Pathologists (CAP) guidelines as ER \>= 1% positive nuclear staining * Completion of all planned cancer treatments prior to registration: * Surgical resection of breast and nodal surgery; (NOTE: Reconstructive surgery does not have to be completed) * Adjuvant radiation therapy, if needed * Neoadjuvant and/or adjuvant chemotherapy, if needed * Post-menopausal defined as * Age \>= 60 and amenorrhea \> 12 consecutive months OR * Previous bilateral oophorectomy OR * Age \< 60 and amenorrhea \> 12 consecutive months and documented follicle stimulating hormone (FSH) level within post-menopausal range according to institutional standard * NOTE: Patients who did not meet these criteria at time of diagnosis and received pre-operative (neoadjuvant) or post-operative (adjuvant) chemotherapy will not be allowed to participate * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, or 2 * Hemoglobin \>= 8.0 g/dL (obtained =\< 14 days prior to registration) * Absolute neutrophil count (ANC) \>= 1500/mm\^3 (obtained =\< 14 days prior to registration) * Platelet count \>= 70,000/mm\^3 (obtained =\< 14 days prior to registration) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (obtained =\< 14 days prior to registration) * Alanine aminotransferase (ALT) and aspartate transaminase (AST) =\< 3 x ULN (obtained =\< 14 days prior to registration) * Ability to swallow oral medication * Provide written informed consent * Willingness to provide mandatory blood specimens for correlative research * Willing to return to enrolling institution for follow-up (during the Active Monitoring Phase of the study) * RE-REGISTRATION-INCLUSION CRITERIA * Confirmation that baseline blood sample was drawn and submitted * Blood estrogen levels after cycle 1 anastrozole (ANA1) must meet the following criteria: * E1 \>= 1.3 pg/ml, AND * E2 \>= 0.5 pg/ml Exclusion Criteria: * REGISTRATION-EXCLUSION CRITERIA * Pre-menopausal women receiving ovarian function suppression (goserelin, leuprolide, etc.) * Stage IV (metastatic) breast cancer * HER2 positive breast cancer as defined by * HER2 immunohistochemistry (IHC) \>= 3+ * HER2/CEP17 \>= 2.0 * HER2/CEP17 \< 2.0 and average HER2 copy number of \>= 6.0 signals/cell * Prior endocrine therapy for this breast cancer. Exceptions: * Pre-operative aromatase therapy (anastrozole, letrozole, or exemestane) and last treatment was \>= 4 weeks prior to registration OR * Pre-operative tamoxifen therapy and last treatment was \>= 12 weeks prior to registration * Currently receiving any of the following cancer-directed therapies: * Radiation therapy * Systemic therapy such as chemotherapy (standard or investigational) * Bisphosphonate therapy started \< 4 weeks prior to registration * NOTE: If patient is currently on bisphosphonate therapy she must be on stable dose for \>= 4 weeks prior to registration. Patients not currently taking bisphosphonates will be allowed to start bisphosphonate therapy after completion of anastrozole (1 mg and 10 mg daily \[if given\]). Information regarding bisphosphonate therapy will be collected * Current use of systemic or topical exogenous estrogen or progesterone (menopausal hormone replacement therapy \[HRT\]) * Prior ovarian function suppression (leuprolide, goserelin, etc.) * Inability to provide informed consent * History of contralateral ductal carcinoma in situ (DCIS) or invasive breast cancer * NOTE: Exception allowed if * Patient did not receive adjuvant endocrine therapy OR * Patient received adjuvant endocrine therapy but has been off treatment for at least 6 months prior to registration * Concurrent active malignancy or history of malignancy =\< 3 years prior to registration * NOTE: Exceptions allowed for successfully treated cervical carcinoma in situ, lobular carcinoma in situ of the breast, papillary thyroid cancer, or non-melanoma skin cancer * Prior prevention therapy with an aromatase inhibitor or a selective estrogen receptor modulator (SERM). Exception: Therapy with a SERM (tamoxifen or raloxifene) is allowed if patient has been off treatment for \>= 6 months prior to registration