Author Summary In recent years, considerable attention has been given to the evolution of drug resistance in the influenza A H1N1 strain. As a major annual cause of morbidity and mortality, combined with the rapid global spread of drug resistance, influenza remains as one of the most important global health concerns. Our work here focuses on a novel multi-faceted population-genetic approach utilizing unique whole-genome multi-time point experimental datasets in both the presence and absence of drug treatment. In addition, we present novel theoretical results and two newly developed and widely applicable statistical methodologies for utilizing time-sampled data - with a focus on distinguishing the relative contribution of genetic drift from that of positive and purifying selection. Results illustrate the available mutational paths to drug resistance, and offer important insights in to the mode and tempo of adaptation in a viral population. The challenge of distinguishing genetic drift from selection remains a central focus of population genetics. Time-sampled data may provide a powerful tool for distinguishing these processes, and we here propose approximate Bayesian, maximum likelihood, and analytical methods for the inference of demography and selection from time course data. Utilizing these novel statistical and computational tools, we evaluate whole-genome datasets of an influenza A H1N1 strain in the presence and absence of oseltamivir (an inhibitor of neuraminidase) collected at thirteen time points. Results reveal a striking consistency amongst the three estimation procedures developed, showing strongly increased selection pressure in the presence of drug treatment. Importantly, these approaches re-identify the known oseltamivir resistance site, successfully validating the approaches used. Enticingly, a number of previously unknown variants have also been identified as being positively selected. Results are interpreted in the light of Fisher's Geometric Model, allowing for a quantification of the increased distance to optimum exerted by the presence of drug, and theoretical predictions regarding the distribution of beneficial fitness effects of contending mutations are empirically tested. Further, given the fit to expectations of the Geometric Model, results suggest the ability to predict certain aspects of viral evolution in response to changing host environments and novel selective pressures.