Narrow-band interference (NBI) and impulsive noise (IN) are two kinds of non-Gaussian noise that have a severe impact on vehicular communications. In this paper, a novel compressive sensing (CS)-based method of simultaneous NBI and IN mitigation is proposed, which is a double kill of the two kinds of unfavorable disturbances. A CS-based time-frequency-measuring orthogonal frequency-division multiplexing (CS-TFM-OFDM) frame structure is introduced, in which the temporal repeated training sequences (TSs) are exploited by the proposed CS-based differential measuring (CS-DM) method to acquire the CS measurement vector of the NBI. With the aid of the a priori partial support, we further proposed the a priori-aided sparsity adaptive matching pursuit (PA-SAMP) to improve the accuracy and stability of NBI recovery. Meanwhile, the measurement vector of the IN is acquired from the null subcarriers in the CS-TFM-OFDM frame. After partial support of the IN is obtained, the IN is reconstructed using the proposed PA-SAMP algorithm. Hence, the two categories of non-Gaussian noise are both thoroughly eliminated, leading to the stability and robustness of vehicular communications. The proposed CS-based approach outperforms the conventional noise suppression methods in vehicular communications environments, which is validated by theoretical analysis and computer simulations.