In this paper we present a computational method for solving delay differential equations of fractional order by employing the Laplace Adomian decomposition method. This method is combined from the Laplace transforms and the Adomian decomposition method taking into account the Caputo derivative as a motivation to describe the fractional derivative. The method is a modification of the Adomian decomposition method and is tested on two examples in order to illustrate the pertinent feature of this method the results shows that the proposed method is an effective and powerful tool for solving delay differential equations of fractional order. A comparison with the exact solution and with the existing methods such as Adomian decomposition method and homotopy analysis method is made.