Coprime arrays have shown potential advantages for direction-of-arrival (DOA) estimation by increasing the number of degrees-of-freedom in the difference coarray domain with fewer physical sensors. In this paper, a new DOA estimation algorithm for coprime array based on the estimation of signal parameter via rotational invariance techniques (ESPRIT) is proposed. We firstly derive the observation vector of the virtual uniform linear array but the covariance matrix of this observation vector is rank-deficient. Different from the traditional Toeplitz matrix reconstruction method using the observation vector, we propose a modified Toeplitz matrix reconstruction method using any non-zero row of the covariance matrix in the virtual uniform linear array. It can be proved in theory that the reconstructed Toeplitz covariance matrix has full rank. Therefore, the improved ESPRIT method can be used for DOA estimation without peak searching. Finally, the closed-form solution for DOA estimation in coprime array is obtained. Compared to the traditional coprime multiple signal classification (MUSIC) methods, the proposed method circumvents the use of spatial smoothing technique, which usually results in performance degradation and heavy computational burden. The effectiveness of the proposed method is demonstrated by numerical examples.