Evolution of antibiotic resistance is a world health crisis, fueled by new mutations. Drugs to slow mutagenesis could, as cotherapies, prolong the shelf-life of antibiotics, yet evolution-slowing drugs and drug targets have been underexplored and ineffective. Here, we used a network-based strategy to identify drugs that block hubs of fluoroquinolone antibiotic-induced mutagenesis. We identify a U.S. Food and Drug Administration- and European Medicines Agency-approved drug, dequalinium chloride (DEQ), that inhibits activation of the Escherichia coli general stress response, which promotes ciprofloxacin-induced (stress-induced) mutagenic DNA break repair. We uncover the step in the pathway inhibited: activation of the upstream "stringent" starvation stress response, and find that DEQ slows evolution without favoring proliferation of DEQ-resistant mutants. Furthermore, we demonstrate stress-induced mutagenesis during mouse infections and its inhibition by DEQ. Our work provides a proof-of-concept strategy for drugs to slow evolution in bacteria and generally.