In terms of the problem that the suppression of impulse noise (IN) in orthogonal frequency division multiplexing (OFDM) based power line communication depends on the characteristic parameters and sparsity of IN, this paper proposed a suppression algorithm based on compressive sensing (CS) and False-Alarm probability. In this approach, the projection of IN is observed using OFDM zero carriers, and the rough estimation of IN is carried out by using the method of base pursuit denoise (BPDN). Then, an adaptive impulse noise detection threshold based on false alarm probability is designed, and the support set of IN is obtained according to the designed threshold. Finally, the IN in power line communication is reconstructed by the least square method. The original OFDM signal can be derived by subtracting the estimated IN from the received OFDM signal. The simulation results show that the proposed algorithm can reconstruct the IN well even if the sparsity conditions of IN is unknown, the performance of the algorithm is superior to the traditional compressed sensing and the nonlinear denoising methods, and this algorithm is reliable and effective for mitigating IN in broadband power line communication.