Direction of arrival (DOA) estimation on uniform linear array with single snapshot has always been a hot topic in radar signal processing. The traditional subspace class estimation methods, such as multiple signal classification (MUSIC), and sparse recovery algorithms, such as subspace pursuit (SP) algorithm, are suitable for different situations, meanwhile have certain requirements for observation data and signal-to-noise ratio (SNR). Combining the advantages of these two methods, a novel MUSIC subspace pursuit (MSP) algorithm is proposed in this paper. Firstly, an error evaluation function of the residual signal is constructed. The angles measured by both methods are replaced and selected one by one. And then, the framework of the greedy algorithm is iteratively applied until convergence. In this algorithm, the subspace information of the signal is added to the greedy algorithm, and more conditions are used to improve the angle measurement accuracy with single snapshot. Simulation results illustrate that the proposed MSP algorithm outperforms the SP and MUSIC algorithms at different SNR levels and it keeps the faster reconstruction speed of SP algorithm.