Various methods have been proposed to rank sports teams. We present here an alternative method that also bases a team’s strength on the strengths of the teams that it defeats, either directly or indirectly. Our Path-Based Method is formulated using a directed graph G whose edges represents the outcomes of each match (win/loss) and with edge-weights based on the number of games played by each team. By only examining simple paths in the graph G, we eliminates the possibility that a team could be (indirectly) credited with defeating itself through cycles in the graph. Unlike other methods, the Path-Based Method does not require arbitrary perturbations to obtain irreducibility or primitivity of the underlying coefficient matrix. Despite using only pairwise comparison information (without regard to home-field advantage or point spread), the method appears to provide an accurate picture of overall team rankings. As a large-scale application, the method is applied to ranking the 128 Division I FBS college football teams at the end of the 2016 season.