Time synchronization is a cooperative work's foundation among underwater sensor networks' nodes; it takes a crucial role in the application and the study of underwater sensor networks. Several time synchronization protocols have been presented for terrestrial radio networks, but they cannot be instantly utilized to real-time underwater sensor networks due to the slow propagation speed of the underwater acoustic signals, the mobility among sensor nodes, and the energy limitation in the underwater wireless sensor networks. In this paper, we propose an adaptive power-efficient time synchronization for mobile underwater sensor networks, called APE-Sync. The proposed scheme takes into account the time-varying clock skew and combines the Doppler-Enhanced synchronization protocol (DE-Sync) and the Kalman filter tracking the clock skew to achieve time synchronization. The proposed scheme also helps in reducing the energy consumption of the nodes by guaranteeing accurate time synchronization. The simulation results are presented, which show that APE-Sync outperforms existent time synchronization schemes in both power efficiency and precision.