We previously described a model of deep intrinsic resistance of breast cancer wherein we used a function-based approach to selection of cancer cells that can survive a variety of challenges in prolonged but reversible quiescence. Our experimental results suggested that resistant cancer cells possess a variety of mechanisms, including modifications of the epigenome and transcriptome, for generating a high degree of cellular heterogeneity. In the present study, we evaluated JIB-04, a small-molecule epigenetic inhibitor initially discovered to inhibit cancer growth, to determine its ability to affect deep intrinsic resistance in our breast cancer model. We found that long pretreatment with JIB-04 sensitized resistant triple-negative inflammatory breast cancer cells and their parental cell line SUM149 to the chemotherapeutic drugs doxorubicin and paclitaxel. Resistant cancer cells derived from another inflammatory breast cancer cell line, FC-IBC02, were considerably more sensitive to JIB-04 than was the parental cell line. Investigating a mechanism of sensitization, we found that JIB-04 exposure increased the expression of PD-L1 in resistant cells, suggesting that JIB-04 may also sensitize resistant breast cancer cells to anti-PD-L1 immune therapy. Finally, these results support the usefulness of our experimental strategy for evaluating anticancer agents such as JIB-04 that may halt cancer evolution and prevent development of cancer resistance to currently used therapies.