Antimicrobial treatments increasingly rely on multidrug combinations, in part because of the emergence and spread of antibiotic resistance. The continued effectiveness of combination treatments depends crucially on the frequency with which multidrug resistance arises. Yet, it is unknown how this propensity for resistance depends on cross-resistance and on epistatic interactions-ranging from synergy to antagonism-between the drugs. Here, we analyzed how interactions between pairs of drugs affect the spontaneous emergence of resistance in the medically important pathogen Staphylococcus aureus. Resistance is selected for within a window of drug concentrations high enough to inhibit wild-type growth but low enough for some resistant mutants to grow. Introducing an experimental method for high-throughput colony imaging, we counted resistant colonies arising across a two-dimensional matrix of drug concentrations for each of three drug pairs. Our data show that these different drug combinations have significantly different impacts on the size of the window of drug concentrations where resistance is selected for. We framed these results in a mathematical model in which the frequencies of resistance to single drugs, cross-resistance, and epistasis combine to determine the propensity for multidrug resistance. The theory suggests that drug pairs which interact synergistically, preferred for their immediate efficacy, may in fact favor the future evolution of resistance. This framework reveals the central role of drug epistasis in the evolution of resistance and points to new strategies for combating the emergence of drug-resistant bacteria.