Identifying and correcting key errors in the final gear-finishing grinding process is crucial for enhancing the accuracy of the finished gears. However, the mechanism by which geometric and thermal errors affect the accuracy of CNC gear profile grinders is unclear, and there is a lack of effective methods for identifying key errors, which complicates the error compensation process. Therefore, this paper proposes an extended Fourier amplitude sensitivity test (EFAST)-based approach for identifying key geometric and thermal errors for gear profile grinders. Firstly, a geometric and thermal error-tooth surface error model (GTE-TSE) using the homogeneous transformation matrix (HTM) is established. Next, an EFAST-based identification method is proposed, which utilizes sensitivity indices to find key errors and their corresponding sensitive components. Finally, the Sobol method and key error correction are employed to perform comparative verification and correctness validation, respectively. The results indicate that after error compensation, the error reduction rate of tooth surface errors (TSEs) reaches 94.51%, proving the crucial impact of identified errors on TSEs and the effectiveness of the proposed method. It provides a theoretical basis for enhancing sensitive components’ accuracy and the follow-up targeted error compensation.