This paper proposed a finite-time prescribed performance control scheme for underactuated autonomous underwater vehicles (AUVs) based on adaptive neural networks and a tan-type barrier Lyapunov function. Even in the presence of output constraints and environmental disturbances, the AUV can also precisely track the desired trajectory in a finite time. By introducing a tan-type barrier Lyapunov Function (TBLF), the singularity problem in process design is solved and all output errors are guaranteed to satisfy the prescribed performance specifications. Dynamic surface control (DSC) and the minimal learning parameter (MLP) are employed to greatly simplify the complexity of the algorithm and enhance the robustness of the control system, respectively. Lyapunov stability analysis proves that the proposed controller guarantees all signals in the closed-loop system to be uniformly ultimately bounded (UUB), and that the tracking errors converge to a small neighborhood near the origin in a finite time. Finally, the simulation results demonstrate the effectiveness and feasibility of the proposed controller.