This paper presents a predictor-based neural dynamic surface control (PNDSC) design method for a class of uncertain nonlinear systems in a strict-feedback form. In contrast to existing NDSC approaches where the tracking errors are commonly used to update neural network weights, a predictor is proposed for every subsystem, and the prediction errors are employed to update the neural adaptation laws. The proposed scheme enables smooth and fast identification of system dynamics without incurring high-frequency oscillations, which are unavoidable using classical NDSC methods. Furthermore, the result is extended to the PNDSC with observer feedback, and its robustness against measurement noise is analyzed. Numerical and experimental results are given to demonstrate the efficacy of the proposed PNDSC architecture.