Genetic perturbations that affect bacterial resistance to antibiotics have been characterized genome-wide, but how do such perturbations interact with subsequent evolutionary adaptation to the drug? Here, we show that strong epistasis between resistance mutations and systematically identified genes can be exploited to control spontaneous resistance evolution. We evolved hundreds of Escherichia coli K-12 mutant populations in parallel, using a robotic platform that tightly controls population size and selection pressure. We find a global diminishing-returns epistasis pattern: strains that are initially more sensitive generally undergo larger resistance gains. However, some gene deletion strains deviate from this general trend and curtail the evolvability of resistance, including deletions of genes for membrane transport, LPS biosynthesis, and chaperones. Deletions of efflux pump genes force evolution on inferior mutational paths, not explored in the wild type, and some of these essentially block resistance evolution. This effect is due to strong negative epistasis with resistance mutations. The identified genes and cellular functions provide potential targets for development of adjuvants that may block spontaneous resistance evolution when combined with antibiotics.