Two distinct strategies for controlling an emerging epidemic are physical distancing and regular testing with self-isolation. These strategies are especially important before effective vaccines or treatments become widely available. The testing strategy has been promoted frequently but used less often than physical distancing to mitigate COVID-19. We compared the performance of these strategies in an integrated epidemiological and economic model that includes a simple representation of transmission by “superspreading,” wherein a relatively small fraction of infected individuals cause a large share of infections. We examined the economic benefits of distancing and testing over a wide range of conditions, including variations in the transmissibility and lethality of the disease meant to encompass the most prominent variants of COVID-19 encountered so far. In a head-to-head comparison using our primary parameter values, both with and without superspreading and a declining marginal value of mortality risk reductions, an optimized testing strategy outperformed an optimized distancing strategy. In a Monte Carlo uncertainty analysis, an optimized policy that combined the two strategies performed better than either one alone in more than 25% of random parameter draws. Insofar as diagnostic tests are sensitive to viral loads, and individuals with high viral loads are more likely to contribute to superspreading events, superspreading enhances the relative performance of testing over distancing in our model. Both strategies performed best at moderate levels of transmissibility, somewhat lower than the transmissibility of the ancestral strain of SARS-CoV-2.