Reliability sensitivity analysis is important to measure how uncertainties influence the reliability of mechanical systems. This study aims to propose an efficient computational method for reliability sensitivity analysis with high accuracy and efficiency. In this study, coordinates of some points on the limit state function are first calculated through Levenberg–Marquardt (LM) iterative algorithm, and the partial derivative of system response relative to uncertain variables is obtained. The coordinate mapping relation and the partial derivative mapping relation are then established by radial basis function neural network (RBFNN) according to these points calculated by the LM iterative algorithm. Following that, the failure samples can be screened out from the Monte Carlo simulation (MCS) sample set by the well-established mapping relations. Finally, the reliability sensitivity is calculated by these failure samples and kernel function, and the failure probability can be obtained correspondingly. Two benchmark examples and an application of industrial robot are used to demonstrate the effectiveness of the proposed method.