A Monte Carlo (MC) scheme for tracking filling fronts is developed. The test problem of tracking fluid injected into a thin mold is considered. The MC scheme iteratively redistributes the volume entering the mold among the cells of a spatial discretization. The transition probabilities used in the MC scheme, which determine how the fluid volume is redistributed, are derived from a discrete representation of the governing steady-state pressure equation. Analysis shows that the MC steps are equivalent to an iterative solution of the discrete equations. Further, it is shown that the MC scheme can be reconfigured into the form of a standard Lattice Boltzmann Method (LBM). Results show that the proposed MC scheme is accurate, does not require an explicit field calculation of the fluid pressure field, and, when compared with existing numerical filling algorithms, exhibits computation times over 1000 times faster.