Myriad experiments have identified an important role for CD8+ T cell response mechanisms in determining recovery from influenza A virus infection. Animal models of influenza infection further implicate multiple elements of the immune response in defining the dynamical characteristics of viral infection. To date, influenza virus models, while capturing particular aspects of the natural infection history, have been unable to reproduce the full gamut of observed viral kinetic behavior in a single coherent framework. Here, we introduce a mathematical model of influenza viral dynamics incorporating innate, humoral, and cellular immune components and explore its properties with a particular emphasis on the role of cellular immunity. Calibrated against a range of murine data, our model is capable of recapitulating observed viral kinetics from a multitude of experiments. Importantly, the model predicts a robust exponential relationship between the level of effector CD8+ T cells and recovery time, whereby recovery time rapidly decreases to a fixed minimum recovery time with an increasing level of effector CD8+ T cells. We find support for this relationship in recent clinical data from influenza A (H7N9) hospitalized patients. The exponential relationship implies that people with a lower level of naive CD8+ T cells may receive significantly more benefit from induction of additional effector CD8+ T cells arising from immunological memory, itself established through either previous viral infection or T cell-based vaccines.