To address the issue of position instability in the strongly coupled and nonlinear cable-drogue-UAV system (CDUS), which arises from rapidly varying disturbances induced by engine shutdown in unmanned aerial vehicles (UAV), this paper proposes a novel trajectory stabilization control strategy that employs a radial basis function neural network (RBFNN)-based backstepping method. Firstly, the affine nonlinear dynamics of CDUS are derived. Then, given the CDUS may be significantly affected by various disturbances, including the tail vortex, atmospheric turbulence, and engine shutdown, an RBFNN observer is constructed to quickly and accurately compensate for the unmeasurable nonlinear dynamics. Therefore, the lumped disturbance approximated by the observer is introduced into the controller design as control compensation, and combined with backstepping control theory, a controller that exhibits high resistance to rapid disturbances is proposed to enhance the trajectory stability of the CDUS. Finally, the flight simulation verifies the proposed method’s effectiveness in enhancing UAV aerial recovery’s stability and response speed.