This paper proposes a finite-time synchronization method based on a Fuzzy Zeroing Neural Network (FZNN) for a class of continuous-time chaotic systems. First, the chaotic system is modeled as a nonlinear system with unknown parameters. Second, the synchronization error between the master and slave chaotic systems is defined. By incorporating a fuzzy control factor, an FZNN controller model with noise suppression capabilities is developed, which ensures that the synchronization process is no longer affected by unknown initial parameter inputs under bounded noise conditions. Subsequently, theoretical analysis demonstrates that the integration of the fuzzy control factor and nonlinear activation functions enables the construction of a finite-time convergent ZNN controller. This controller achieves exact synchronization of the master-slave systems within a predefined time that can be set arbitrarily, exhibiting strong robustness against initial conditions and external noise. Furthermore, numerical examples involving two chaotic systems are provided to verify the effectiveness of the proposed scheme. Finally, the FZNN-based synchronization scheme is applied to image encryption and decryption, validating the feasibility of the method in practical applications.