Crosstalk and complexity within signaling pathways and their perturbation by oncogenes limit component-by-component approaches to understanding human disease. Network analysis of how normal and oncogenic signaling can be rewired by drugs may provide opportunities to target tumors with high specificity and efficacy. Using targeted inhibition of oncogenic signaling pathways, combined with DNA-damaging chemotherapy, we report that time-staggered EGFR inhibition, but not simultaneous coadministration, dramatically sensitizes a subset of triple-negative breast cancer cells to genotoxic drugs. Systems-level analysis—using high-density time-dependent measurements of signaling networks, gene expression profiles, and cell phenotypic responses in combination with mathematical modeling—revealed an approach for altering the intrinsic state of the cell through dynamic rewiring of oncogenic signaling pathways. This process converts these cells to a less tumorigenic state that is more susceptible to DNA damage-induced cell death by reactivation of an extrinsic apoptotic pathway whose function is suppressed in the oncogene-addicted state.