The growing field of spintronics relies on new techniques and technologies for injecting and detecting electron spins to generate spin-dependent signals and utilize spin as a new state variable. Magnetic tunnel junctions (MTJs) do this by employing thin oxide layers as insulating barriers between two ferromagnetic metals, but the oxides suffer from defects and material interdiffusion that limit device performance. In this work, we demonstrate that graphene, a material widely studied for its high lateral conductance, functions as a tunnel barrier in the out-of-plane direction. We fabricate graphene-based MTJs and characterize spin and charge transport as a function of bias and temperature from 4 to 425 K. The device behavior fits well with traditional charge and spin-polarized tunneling transport models. This result has implications for development of new, ultra-low power spin-based devices such as magnetic random access memory (MRAM), spin logic, and reconfigurable circuits.