We propose a Taylor expansion multiple signal classification (TE MUSIC) method for joint direction of departure (DOD) and direction of arrival (DOA) estimation in a bistatic multiple-input multiple-output (MIMO) array. First, using a Taylor expansion of the steering vector, a two-dimensional (2D) search in the conventional MUSIC method for MIMO arrays is reduced to a two-step one-dimensional (1D) search in the proposed TE MUSIC method. Second, DOAs of the targets can be achieved via Lagrange multiplier by a 1D search. Finally, substituting the DOA estimates into the 2D MUSIC spectrum function, DODs of the targets are obtained by another 1D search. Thus, the DOD and DOA estimates can be automatically paired. The performance of the proposed method is better than that of the MIMO ESPRIT method, and is similar to that of the 2D MUSIC method. Furthermore, due to the 1D search, the TE MUSIC method avoids the high computational complexity of the 2D MUSIC method. Simulation results are presented to show the effectiveness of the proposed method.