Uncertain disturbances in three-phase PWM rectifiers, such as load disturbances and grid voltage fluctuations, cause significant fluctuations in the DC-link voltage and even lead to system instability. To enhance the disturbance-rejection ability, this paper establishes a dynamic model of the three-phase rectifier considering system disturbances in the two-phase stationary coordinate frame, and proposes a fixed-time adaptive neural network control (FTANNC) method based on a cascaded dual-loop structure. In the outer dc-link voltage loop, an adaptive neural network approach is employed to estimate for uncertainties such as load disturbances, while a voltage controller is constructed to ensure fast and accurate voltage regulation. In the inner power loop, a new command filter is used to estimate for the derivative of the power reference, and a fixed-time power controller is designed to guarantee the power tracking performance. Theoretical analysis and test results validate that the proposed method possesses fast dynamic response and good disturbance rejection capability. Compared with existing approaches, such as PI with feedforward control, active disturbance rejection control, and DPC-BSC, the FTANNC method reduces DC voltage settling time by nearly 76.5% and steady-state tracking error by approximately 36.8% under voltage regulation. Under load disturbances and grid voltage fluctuations, it shortens settling time by at least 40% while minimizing voltage variations.