Direction-of-arrival (DOA) estimation with a co-prime linear array, composed of two uniform linear arrays with inter-element spacing larger than half-wavelength of incoming signals, has been investigated a lot thanks to its high-resolution performance. For better computational efficiency, one class of methods treat the co-prime linear array as two sparse uniform linear subarrays. From each of them, high-precision but ambiguous DOA estimation is obtained, and the ambiguities are eliminated according to the co-prime property. However, the existing methods of this kind suffer from the insufficient reliability and high complexity. In this paper, the potential problems associated with the DOA estimation with two co-prime subarrays are discussed, and a reliable and efficient DOA estimation method is proposed. For each subarray, the true DOAs are treated as their equivalent angles and the pair matching of them is accomplished by exploring the cross-correlations between the equivalent signals associated with the equivalent angles. Compared with other existing methods, the proposed method is able to achieve a better estimation performance in all situations, in terms of accuracy and complexity.