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A coprime array has a larger array aperture as well as increased degrees-of-freedom (DOFs), compared with a uniform linear array with the same number of physical sensors. Therefore, in a practical wireless communication system, it is capable to provide desirable performance with a low-computational complexity. In this study, the authors focus on the problem of efficient direction-of-arrival (DOA) estimation, where a coprime array is incorporated with the idea of compressive sensing. Specifically, the authors first generate a random compressive sensing kernel to compress the received signals of coprime array to lower-dimensional measurements, which can be viewed as a sketch of the original received signals. The compressed measurements are subsequently utilised to perform high-resolution DOA estimation, where the large array aperture of the coprime array is maintained. Moreover, the authors also utilise the derived equivalent virtual array signal of the compressed measurements for DOA estimation, where the superiority of coprime array in achieving a higher number of DOFs can be retained. Theoretical analyses and simulation results verify the effectiveness of the proposed methods in terms of computational complexity, resolution, and the number of DOFs.