Conventional compressive sensing (CS)-based imaging methods allow images to be reconstructed from a small amount of data, while they suffer from high computational burden even for a moderate scene. To address this problem, this paper presents a novel two-dimensional (2D) CS imaging algorithm for strip-map synthetic aperture radars (SARs) with zero squint angle. By introducing a 2D separable formulation to model the physical procedure of the SAR imaging, we separate the large measurement matrix into two small ones, and then the induced algorithm can deal with 2D signal directly instead of converting it into 1D vector. As a result, the computational load can be reduced significantly. Furthermore, thanks to its superior performance in maintaining contour information, the gradient space of the SAR image is exploited and the total variation (TV) constraint is incorporated to improve resolution performance. Due to the non-differentiable property of the TV regularizer, it is difficult to directly solve the induced TV regularization problem. To overcome this problem, an improved split Bregman method is presented by formulating the TV minimization problem into a sequence of unconstrained optimization problem and Bregman updates. It yields an accurate and simple solution. Finally, the synthesis and real experiment results demonstrate that the proposed algorithm remains competitive in terms of high resolution and high computational efficiency.