An integral predictor-based dynamic surface control scheme is developed with prescribed performance (IPPDSC) for multi-motor driving servo systems in this paper. By employing a novel finite-time performance function and an improved error transformation, the tracking error is limited within a prescribed zone in any preset time without having the overrun and the singularity problem. Furthermore, integral state predictors are designed to update neural network weights to handle high-frequency oscillations under large adaptive gains. Different from the existing approaches, an integral term of prediction error is introduced to eliminate the steady-state error and avoid chattering. In addition, a synchronization controller based on the mean relative coupling structure is proposed to solve the coupling problem between synchronization and tracking. Finally, simulation and experimental results are presented to demonstrate the effectiveness of the designed approach.