This paper presents a robust neural network with an extended state observer control methodology for a piezoelectric-actuator-based surgical device. This control methodology is proposed for tracking of desired motion trajectories in the presence of unknown or uncertain system parameters, nonlinearities including friction, hysteresis, and disturbances in the motion system. In particular, the radial basis function neural network, which serves as a function approximator, aims to create a model for an unknown function to find a relationship between input and output data. An extended state observer is utilized to assist in canceling disturbances and uncertainties of the system dynamically. The stability of the control approach is analyzed. The convergence of position and velocity tracking errors is proven theoretically. Experiments are conducted to demonstrate that the performance with an improved accuracy can be attained by the proposed control scheme. With the motion tracking capability, the control methodology helps the novel surgical device achieve higher success rate in operation, which is also suitable for similar piezoelectric ultrasonic actuator applications.