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Many cancers harbor pro-proliferative mutations of the mitogen-activated protein kinase (MAPK) pathway. In BRAF-driven melanoma cells treated with BRAF inhibitors, subpopulations of cells escape drug-induced quiescence through a nongenetic manner of adaptation and resume slow proliferation. Here, we found that this phenomenon is common to many cancer types driven by EGFR, KRAS, or BRAF mutations in response to multiple, clinically approved MAPK pathway inhibitors. In 2D cultures and 3D spheroid models of various cancer cell lines, a subset of cells escaped drug-induced quiescence within 4 days to resume proliferation. These “escapee” cells exhibited DNA replication deficits, accumulated DNA lesions, and mounted a stress response that depended on the ataxia telangiectasia and RAD3-related (ATR) kinase. We further identified that components of the Fanconi anemia (FA) DNA repair pathway are recruited to sites of mitotic DNA synthesis (MiDAS) in escapee cells, enabling successful completion of cell division. Analysis of patient tumor samples and clinical data correlated disease progression with an increase in DNA replication stress response factors. Our findings suggest that many MAPK pathway–mutant cancers rapidly escape drug action and that suppressing early stress tolerance pathways may achieve more durable clinical responses to MAPK pathway inhibitors.