Standard Direction of Arrival (DOA) estimation methods are typically derived based on the Gaussian noise assumption, making them highly sensitive to outliers. Therefore, in the presence of impulsive noise, the performance of these methods may significantly deteriorate. In this paper, we model impulsive noise as Gaussian noise mixed with sparse outliers. By exploiting their statistical differences, we propose a novel DOA estimation method based on sparse signal recovery (SSR). Furthermore, to address the issue of grid mismatch, we utilize an alternating optimization approach that relies on the estimated outlier matrix and the on-grid DOA estimates to obtain the off-grid DOA estimates. Simulation results demonstrate that the proposed method exhibits robustness against large outliers.