Influenza virus inflicts a heavy death toll annually and resistance to existing antiviral drugs has generated interest in the development of agents with novel mechanisms of action. Favipiravir is an antiviral drug that acts by increasing the genome-wide mutation rate of influenza A virus (IAV). Potential synergistic benefits of combining oseltamivir and favipiravir have been demonstrated in animal models of influenza, but the population-level effects of combining the drugs are unknown. In order to elucidate the underlying evolutionary processes at play, we performed genome-wide sequencing of IAV experimental populations subjected to serial passaging in vitro under a combined protocol of oseltamivir and favipiravir. We describe the interplay between mutation, selection, and genetic drift that ultimately culminates in population extinction. In particular, selective sweeps around oseltamivir resistance mutations reduce genome-wide variation while deleterious mutations hitchhike to fixation given the increased mutational load generated by favipiravir. This latter effect reduces viral fitness and accelerates extinction compared with IAV populations treated with favipiravir alone, but risks spreading both established and newly emerging mutations, including possible drug resistance mutations, if transmission occurs before the viral populations are eradicated.