In this article, the problems of fault diagnosis (FD) and fault-tolerant control (FTC) are investigated for a class of nonlinear systems with output measurement noise. Due to the influence of measurement noise in the output sensor, the output observation error cannot be accurately obtained, which causes obstacles to the accuracy of FD. To address this issue, an output filter and disturbance estimator are constructed to decrease the negative effects of measurement noise and observer gain disturbances, and a novel non-fragile neural observer is designed to estimate the unknown states. A new evaluation function is also introduced to detect faults. Then, a novel neural FTC controller is proposed in the presence of faults, to ensure that all the closed-loop system signals are semiglobally uniformly ultimately bounded (SGUUB). The effectiveness of the proposed methodology is verified via numerical simulation of a one-link robot system.