This paper mainly addresses the model-parameter-free prescribed time trajectory tracking control issue for under-actuated unmanned surface vehicles (USVs) that are susceptible to model uncertainties, time-varying disturbances, and saturation constraints. Firstly, a state extension based on coordinate transformation was designed to address the lack of control in the sway channel. Secondly, nonlinear behavior stemming from saturation constraints is not always differentiable. Regarding this, a smooth dead-zone-based model was conducted to fit the behavior, leaving a relatively simple actuator model. Then, an improved prescribed time–prescribed performance function (PTPPF) and error transformation method were utilized to propose a model-parameter-free control algorithm that guarantees user-defined constrained boundaries while ensuring all tracking errors converge within small domains before a preassigned settling time. The theoretical analysis was conducted by the initial value theorem, Lyapunov’s second method, and proof by contradiction, followed by comparative simulation results that verified the effectiveness of the proposed control scheme.