The rapid evolution of drug resistance remains a critical public health concern. The treatment of influenza A virus (IAV) has proven particularly challenging, due to the ability of the virus to develop resistance against current antivirals and vaccines. Here, we evaluate a novel antiviral drug therapy, favipiravir, for which the mechanism of action in IAV involves an interaction with the viral RNA-dependent RNA polymerase resulting in an effective increase in the viral mutation rate. We used an experimental evolution framework, combined with novel population genetic method development for inference from time-sampled data, to evaluate the effectiveness of favipiravir against IAV. Evaluating whole genome polymorphism data across 15 time points under multiple drug concentrations and in controls, we present the first evidence for the ability of IAV populations to effectively adapt to low concentrations of favipiravir. In contrast, under high concentrations, we observe population extinction, indicative of mutational meltdown. We discuss the observed dynamics with respect to the evolutionary forces at play and emphasize the utility of evolutionary theory to inform drug development.