Let G be a simple, finite, undirected, and connected graph, let [k] = {1,..., k} be the set of positive integers from 1 to k, where k ≥ 1, and let f: E(G) → [k] be a proper edge coloring of G. Then f is said to be a dominator k-edge coloring of G if for each edge e of G, e is uniquely colored or e is adjacent to every edge of some color class. The dominator edge game coloring DEG (G, [k]) of G is a game played by two players, Neil and Bern, who take alternate turns (Neil moves first) coloring an uncolored edge of G with a color in [k] such that the resulting partial coloring is proper. The game ends when all edges colored or when a player cannot make a feasible move. We say that G is dominator k-edge game colorable if Neil has a winning strategy in DEG (G, [k]) that guarantees that all edges of G will be colored and the resulting edge coloring is a dominator k-edge coloring of G, regardless of Bern\'s optimal counter moves. The minimum value of k for which G is dominator k-edge game colorable is called the dominator game chromatic index of G, and is denoted by Xág (G). In this paper, we investigate the dominator edge game colorability of graphs. In particular, we will determine the dominator game chromatic indices of well- known graphs such as paths, cycles, stars, and complete graphs.