Quantifying adaptive evolution at the genomic scale is an essential yet challenging aspect of evolutionary biology. Here, we develop a method that extends and generalizes previous approaches to estimate the rate of genomic adaptation in rapidly evolving populations and apply it to a large data set of complete human influenza A virus genome sequences. In accord with previous studies, we observe particularly high rates of adaptive evolution in domain 1 of the viral hemagglutinin (HA1). However, our novel approach also reveals previously unseen adaptation in other viral genes. Notably, we find that the rate of adaptation (per codon per year) is higher in surface residues of the viral neuraminidase than in HA1, indicating strong antibody-mediated selection on the former. We also observed high rates of adaptive evolution in several nonstructural proteins, which may relate to viral evasion of T-cell and innate immune responses. Furthermore, our analysis provides strong quantitative support for the hypothesis that human H1N1 influenza experiences weaker antigenic selection than H3N2. As well as shedding new light on the dynamics and determinants of positive Darwinian selection in influenza viruses, the approach introduced here is applicable to other pathogens for which densely sampled genome sequences are available, and hence is ideally suited to the interpretation of next-generation genome sequencing data.