An alternating efficient approach for predicting non-stationary response of randomly excited nonlinear systems is proposed by a combination of radial basis function neural network (RBFNN) and stochastic averaging method (SAM). First, the n-degree-of-freedom quasi-non-integrable-Hamiltonian (QNIH) system is reduced to a one-dimensional averaged Itô differential equation within the framework of SAM for QNIH. Subsequently, the associated Fokker–Planck–Kolmogorov (FPK) equation is solved with the RBFNN. Specifically, the solution of the associated FPK equation is expressed in a linear combination of a series of basis functions with time-correlation weights. These time-depended weights are solved by minimizing a loss function, which involves the residual of the differential equations and the constraint conditions. Three typical nonlinear systems are studied to verify the applicability of the developed scheme. Comparisons to the data generated by simulation technique indicate that the approach yields reliable results with high efficiency.