The trajectory tracking of the autonomous surface vessel (ASV) can significantly benefit the ocean exploration in terms of advanced state estimation, accurate motion control, etc. This work focuses on the finite-time asymptotic trajectory tracking control issue for the ASV subject to saturation nonlinearities, model uncertainties, and external disturbances. First, to estimate the yaw rate of the ASV, an adaptive cubature Kalman filter (ACKF) estimator is designed, in which an adaptive dynamic programming is utilized to tackle the model mismatches and generate a compensation term for the reduction of the estimation error. Second, the saturation model with smooth function is presented to deal with the saturation nonlinearities, aiming to restrict the surge force, sway force, and yaw moment of the ASV in the predefined boundaries. Third, a finite-time prescribed performance robust integral of the sign of the error (RISE) control method with the radial basis function neural network is proposed to overcome model uncertainties and external disturbances, dedicating to guarantee the transient performance and achieve the asymptotic trajectory tracking of the ASV. For the closed-loop system, the finite-time asymptotic stability is proved by employing the Lyapunov stability theorem. The effectiveness of the proposed estimator-based control method is validated by numerical simulations.