A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present a polynomial-time algorithm that uses O(g^{1/2} n^{1/2} log n)-space to find an O(g^{1/2} n^{1/2})-sized separator of a graph having n vertices and embedded on an orientable surface of genus g.