Due to the combined influence of system randomness and strong nonlinearity, the dynamic response analysis of nonlinear random parameter systems is very challenging. To solve this problem, a new analytical approach based on stochastic perturbation-Galerkin method is proposed, and the dynamic response of single-degree-of-freedom nonlinear stochastic systems is studied. Combined with the higher-order perturbation and Newmark integral scheme, the dynamic response of the system is obtained by power series expansion. Then a new series expansion is obtained by using the Galerkin projection with the expansion terms of power series of different orders as trial functions. This projection method ensures the statistical minimization of stochastic errors of series approximation. Numerical examples show that the proposed method is more accurate than the perturbation method of the same order, and can maintain better convergence in the long integration process, and has higher efficiency than the direct Monte Carlo simulation method.