To overcome disadvantages of conventional Doppler estimation and compensation on UUV (Unmanned Underwater Vehicle) platform, a novel algorithm based on joint time-frequency searching is proposed to track and rapidly compensate the time-variant Doppler in underwater acoustic OFDM (Orthogonal Frequency Division Multiplexing) communication. The algorithm utilizes the comb pilots combined with frequency-domain resampling to calculate the channel impulse response with different Doppler factors. Then the correct Doppler factor is achieved and eliminated by seeking the sparest channel impulse response, which is accessed using matching pursuit method. Experiment results performed by Monte Carlo simulation and sea-trial data are provided, demonstrating that the proposed algorithm can effectively track and rapidly compensate the time-variant Doppler factor, thus ensuring the stable underwater acoustic OFDM mobile communication for UUV.