Few questions on infectious disease are more important than understanding how and why avian influenza A viruses successfully emerge in mammalian populations, yet little is known about the rate and nature of the virus’ genetic adaptation in new hosts. Here, we measure, for the first time, the genomic rate of adaptive evolution of swine influenza viruses (SwIV) that originated in birds. By using a curated dataset of more than 24 000 human and swine influenza gene sequences, including 41 newly characterized genomes, we reconstructed the adaptive dynamics of three major SwIV lineages (Eurasian, EA; classical swine, CS; triple reassortant, TR). We found that, following the transfer of the EA lineage from birds to swine in the late 1970s, EA virus genes have undergone substantially faster adaptive evolution than those of the CS lineage, which had circulated among swine for decades. Further, the adaptation rates of the EA lineage antigenic haemagglutinin and neuraminidase genes were unexpectedly high and similar to those observed in human influenza A. We show that the successful establishment of avian influenza viruses in swine is associated with raised adaptive evolution across the entire genome for many years after zoonosis, reflecting the contribution of multiple mutations to the coordinated optimization of viral fitness in a new environment. This dynamics is replicated independently in the polymerase genes of the TR lineage, which established in swine following separate transmission from non-swine hosts.