In recent decades, compressive sensing (CS) is a popular theory for studying the inverse problem, and has been widely used in synthetic aperture radar (SAR) image processing. However, the computation complexity of CS-based methods limits its wide applications in SAR imaging. In this paper, we propose a novel sparse SAR imaging method using the Multiple Measurement Vectors model to reduce the computation cost and enhance the imaging result. Based on using the structure information and the matched filter processing, the new CS-SAR imaging method can be applied to high-quality and high-resolution imaging under sub-Nyquist rate sampling with the advantages of saving the computational cost substantially both in time and memory. The results of simulations and real SAR data experiments suggest that the proposed method can realize SAR imaging effectively and efficiently.