To address the model uncertainties and external disturbances of small unmanned underwater vehicles (UUVs) with velocity-free in the trajectory tracking, a robust finite-time prescribed-performance asymptotic tracking control scheme is proposed. Firstly, a novel prescribed performance control is developed by constructing finite-time performance function and error mapping function, where the settling time can be chosen in advance and no longer depends on initial conditions. On this basis, combining the extended state observer (ESO) and robust integral of the sign of error (RISE), an output-feedback controller is proposed to robustness against with lumped disturbances, where the linear ESO is employed to quickly compensate the unknown velocity and lumped disturbances, and the continuous RISE term is designed to further suppress observed errors and avoid chattering. This design not only guarantees the transient and steady-state performance, but also avoids a conservative assumption on the requirement for the precise upper bound of differentials as well as the dependence on high control gains of RISE. The stability analysis proves that all the closed-loop control signals asymptotically converge in finite time. Simulation results verify the effectiveness of the proposed control method.