This paper proposes a new computationally efficient method for two-dimensional (2-D) direction-of-arrivals (DOAs) estimation of multiple correlated narrowband signals impinging on an L-shaped sensor array structured by two uniform linear arrays (ULAs). In the proposed method, the azimuth and elevation angles are estimated independently from the cross-correlation matrices between the received data of two ULA subarrays by linear operation, and then the pair-matching of estimated azimuth and elevation angles can be accomplished by using the correlation matrix of the combined received data of these subarrays. Since the eigen-decomposition is avoided, the proposed DOA estimation method has less computational complexity and better estimation performance than the existing methods. Furthermore the asymptotic mean-square-error (MSE) expressions of the estimation errors are derived, and the effectiveness of the proposed method is verified through numerical examples.