This work presents a new conservative finite-volume numerical solution for the two-dimensional groundwater flow (Boussinesq) equation, which can be used for investigations of hillslope subsurface flow processes and simulations of catchment hydrology. The Boussinesq equation, which is integrated for each grid element, can take account of the local variations of topography and soil properties within the individual elements. The numerical method allows for wetting and drying of the water table, without “ad hoc assumptions.” The stability and convergence of the method is shown to be guaranteed a priori by the properties of the solver itself, even with respect to the boundary conditions, an aspect that has been neglected in previous literature. The numeric solutions are validated against some approximate analytical solutions and compared to those of another (1-D) numerical solver of the Boussinesq equation. The solver capabilities are further explored with simulations of the Panola experimental hillslope where the bedrock topography, which is accurately known, causes complex wetting and drying patterns; in this situation, the importance of a two-dimensional description of subsurface flows to obtain properly simulated discharges becomes clear. Finally, a comparison is made between the results of the presented algorithm and the output of the GEOtop hydrological distributed model, which simulates variably saturated soils; the findings of the comparison are discussed.