Onalespib and Paclitaxel in Treating Patients With Advanced Triple Negative Breast Cancer

Inclusion Criteria: * Patients must have histologically confirmed measurable or unmeasurable advanced or metastatic breast cancer for which standard curative measures do not exist or are no longer effective * Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Primary and/or metastatic breast tumor must be negative for over-expression of estrogen and progesterone receptors; patients with weak estrogen receptor and/or progesterone receptor expression (\< 10% on immunohistochemistry \[IHC\]) will be eligible * Primary and/or metastatic breast tumor must be negative for human epidermal growth factor receptor (HER-2/neu) over-expression based on immunohistochemistry (IHC) (0 or 1+, 2+ if fluorescence in-situ hybridization \[FISH\] test is negative) or FISH (HER2/copy number of centromere of chromosome 17 \[CEP17\] ratio \< 2.0 or \< 4 Her-2/neu signals per nucleus) * Any number of prior therapies for metastatic breast cancer is allowed; patients with weakly estrogen receptor positive breast cancer who received any number of endocrine agents for metastatic breast cancer will also be eligible * Prior taxane is allowed (as long as the patient is not experiencing grade \> 1 neuropathy and had no history of disease progression on a taxane therapy within 3 months prior to study enrollment) * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Life expectancy of greater than 12 weeks * Leukocytes \>= 2,000/uL * Absolute neutrophil count \>= 1,500/uL * Platelets \>= 100,000/uL * Total bilirubin less than or equal to the institution's upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal (except for patients with liver metastases in whom AST/ALT can be \< 5 x institutional upper limit of normal) * Creatinine within normal institutional limits OR creatinine clearance \>= 50 mL/min for patients with creatinine levels above institutional normal * Left ventricular ejection fraction of \> 50% on baseline echocardiography or multi-gated acquisition (MUGA) scan * Corrected QT interval (QTc) of \< 480 milliseconds * Female subjects with child bearing potential must have a negative pregnancy test at screening; child bearing potential is defined as sexually active patients with menses less than 1 year prior to enrollment, \< 65 years of age, have no history of oophorectomy or hysterectomy * The effects of AT13387 on the developing human fetus are unknown; for this reason and because paclitaxel are known to be teratogenic; women of child-bearing potential and men must agree to use adequate contraception prior to study entry, for the duration of study participation and 3 months after completion of study treatment administration; adequate contraception includes methods such as oral contraceptives, double barrier method (condom plus spermicide or diaphragm), or abstaining from sexual intercourse; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients who are receiving any other investigational agents within 4 weeks or 5 half-lives (whichever is shorter) prior to the first dose of the study regimen * Prior radiation therapy within 2 weeks prior to the first dose of the study regimen * Patients in whom prior treatment related toxicities have not recovered to grade 1 or less (except for alopecia) * Recent initiation of bone modifying therapy with a bisphosphonate or denosumab unless it has been started more than 4 weeks prior to the first dose of the study regimen; patients who are already enrolled in this study can initiate bone modifying therapy after the first set of re-staging scans (\>= 8 weeks from cycle 1, day 1) * Prior therapy with AT13387 or another HSP90 inhibitor * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; however, patients with previously treated and stable brain metastases are eligible as long as they are no longer requiring steroids, completed radiation therapy more than 2 weeks prior to the first dose of study regimen and have no seizures or worsening neurologic symptoms * History of grade 3-4 immediate hypersensitivity reaction to paclitaxel * History of clinically significant allergic reactions attributed to compounds of similar chemical or biologic composition to AT13387 or paclitaxel * Based on investigator's brochure, AT13387 has no significant effects on inhibition or activation of cytochrome P450 (CYP), including 1A2, 3A4, 2D6, 2C9, and 2C19 at the half maximal inhibitory concentration (IC50) \> 10 uM. Preclinical studies indicated that AT13387 is only a modest inhibitor of P-glycoprotein (P-gp). Paclitaxel is a substrate of CYP2C8 and CYP3A4. The use of CYP2C8 and CYP3A4 inhibitors/inducers while not prohibited in this study, is discouraged whenever feasible; concurrent use of strong CYP2C8 and CYP3A4 inhibitors/inducers should be documented and the principal investigator (PI) of the study shall be notified prior to dosing; as part of the enrollment/informed consent procedures, the patients will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * 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 are excluded from this study because paclitaxel is a class D agent with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with AT13387 and paclitaxel, breastfeeding should be discontinued if the mother is treated with AT13387 and paclitaxel * Patients who are human immunodeficiency virus (HIV) positive on highly active anti-retroviral therapy (HAART) will be excluded from the study because of the potential for pharmacokinetic interactions with AT13387; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated * Inability to understand and sign informed consent * Any other medical or psychiatric condition that in the opinion of the investigator would make the study therapy unsafe for the patient