We present a new bio-inspired control strategy for an autonomous underwater vehicle by constructing coupled nonlinear oscillators, similar to the animal central pattern generators (CPGs). Using contraction theory, we show that the network of oscillators globally converges to a specific pattern of oscillation. We experimentally validate the proposed control law using a turtle-like underwater vehicle, whose fin actuators successfully exhibit a pattern that resembles the motion of fore limbs of a swimming sea turtle. In order to further fulfill the potential of the CPG-based control, we propose to feed back the actuator states to the coupled oscillators, thereby achieving not only the synchronization of the oscillators, but also the synchronization of actual foil states. Such a closed-loop version of CPGs makes the controller more robust and practical in the presence of external disturbances.