Recent observations of significant enhancements in exospheric hydrogen (H) emission in response to geomagnetic storms have been difficult to interpret in terms of the evolution of the underlying global, 3-D exospheric structure. In this letter, we report the first measurement of the timescales and spatial gradients associated with the exospheric response to a geomagnetic storm, which we derive from a novel, time-dependent tomographic analysis of H emission data. We find that global H density at 3 RE begins to rise promptly, by ∼15%, after storm onset and that this perturbation appears to propagate outward with an effective speed of ∼60 m/s, a response that may be associated with enhanced thermospheric temperature and vertical neutral wind. This effective upwelling has significant implications for atmospheric escape as well as for charge exchange reaction rates, which drive important space weather effects such as plasmaspheric refilling and ring current decay.