McCague replaces Ian Scott who will resume practice within the firm as a partner in the industrial and commercial property department.
McCague has been with Arthur Cox for ten years. The 38-partner firm has offices in Dublin, Belfast and New York.
Chairman, James O'Dwyer says: "Eugene is taking over from Ian at a stage when the firm is recognised as the market leader in key specialised areas such as mergers and acquisitions, corporate finance and foreign investment.
"Our intention is to build on these strengths and to match the speed and direction of our development with the evolution of our clients' needs.