Lens-free digital in-line holography (LDIH) is a promising technology for portable, wide field-of-view imaging. Its resolution, however, is limited by the inherent pixel size of an imaging device. Here we present a new computational approach to achieve sub-pixel resolution for LDIH. The developed method is a sparsity-based reconstruction with the capability to handle the non-linear nature of LDIH. We systematically characterized the algorithm through simulation and LDIH imaging studies. The method achieved the spatial resolution down to one-third of the pixel size, while requiring only single-frame imaging without any hardware modifications. This new approach can be used as a general framework to enhance the resolution in nonlinear holographic systems.