Triple-negative breast cancer (TNBC) is associated with poor prognosis and limited treatment options due to the lack of important receptors (estrogen receptor [ER], progesterone receptor [PR], and human epidermal growth factor receptor 2 [HER2]) used for targeted therapy. However, high-throughput in vitro drug screening of cell lines is a powerful tool for identifying effective drugs for a disease. Here, we determine the intrinsic chemosensitivity of TNBC cell lines to proteasome inhibitors (PIs), thereby identifying potentially potent 2-drug combinations for TNBC. Eight TNBC cell lines (BT-549, CAL-148, HCC1806, HCC38, HCC70, MDA-MB-436, MDA-MB-453, and MDA-MB-468) and two controls (MCF-10A and MCF-7) were first exposed to 18 drugs (11 PIs and 7 clinically relevant chemotherapeutic agents) as monotherapy, followed by prediction of potent 2-drug combinations using the IDACombo pipeline. The synergistic effects of the 2-drug combinations were evaluated with SynergyFinder in four TNBC cell lines (CAL-148, HCC1806, HCC38, and MDA-MB-468) and three controls (BT-474, MCF-7, and T47D) in vitro, followed by further evaluation of tumor regression in zebrafish tumor models established using HCC1806 and MCF-7 cells. Monotherapy identified nine effective drugs (bortezomib, carfilzomib, cisplatin, delanzomib, docetaxel, epoxomicin, MLN-2238, MLN-9708, and nedaplatin) across all cell lines. PIs (e.g., bortezomib, delanzomib, and epoxomicin) were highly potent drugs in TNBC cells, of which bortezomib and delanzomib inhibited the chymotrypsin-like activity of the 20 S proteasome by 100% at 10 µM. Moreover, several potent 2-drug combinations (e.g., bortezomib+nedaplatin and epoxomicin+epirubicin) that killed virtually 100% of cells were also identified. Although HCC1806- and MCF-7-derived xenografts treated with bortezomib+nedaplatin and carboplatin+paclitaxel were smaller, HCC1806 cells frequently metastasized to the trunk region. Taken together, we show that PIs used in combination with platinum agents or topoisomerase inhibitors exhibit increased efficiency with almost 100% inhibition in TNBC cell lines, indicating that PIs are therefore promising compounds to use as combination therapy for TNBC.