In this article, a bistatic multiple-input–multiple-output (MIMO) radar that exploits compressive measurements is proposed, where the compressive sampling is involved in each branch of the receive antenna. To fully utilize the bistatic MIMO radar structure, we develop a high-order singular value decomposition (HOSVD)-based algorithm for the direction-of-departure (DOD) and direction-of-arrival (DOA) joint estimation. It is worth noting that HOSVD divides the DOD and DOA into two independent dimensions, thus generating cross items. Consequently, it is difficult to obtain the exact DOD and DOA results. To address this issue, a pairing algorithm is introduced in this article to connect the DOD and DOA parameters, where the relationship between the covariance tensor and covariance matrix is utilized. Furthermore, the Cramér–Rao bound for the proposed compressive bistatic MIMO radar is derived, and computational complexity is also analyzed from the perspective of received signal dimension. Numerical simulations verify that the proposed scheme outperforms the conventional MIMO radar, and the proposed tensor-based algorithm has a better estimation performance than the matrix-based MUSIC.