In this paper we present an algorithm for finding a minimum dominator coloring of orientations of paths. To date this is the first algorithm for dominator colorings of digraphs in any capacity. We prove that the algorithm always provides a minimum dominator coloring of an oriented path and show that it runs in $\mathcal{O}(n)$ time. The algorithm is available at https://github.com/cat-astrophic/MDC-orientations_of_paths/.