Stochastic response prediction is one of the principal topics in diverse branches of engineering, such as aerospace engineering, mechanical engineering, and vehicle engineering. However, the challenge of addressing high-dimensional partial differential equations or evaluating high-dimensional domain integrals in determining averaged derivative moments has limited the research. This study proposes a method to predict the stochastic response of high-degree-of-freedom (DOF) quasi-non-integrable Hamiltonian systems under Poisson white noises. The original finite DOF system is reduced to a one-dimensional one governing the Hamiltonian by applying the stochastic averaging method, and the associated reduced generalized Fokker–Planck–Kolmogorov (GFPK) equation is derived. Then, the two-step generalized elliptical coordinate transformation is extended to evaluate averaged derivative moments that are the key to obtaining the stationary probability density function (SPDF) for probabilistic information prediction. Moreover, the perturbation method is introduced to address the high-order GFPK equation. Finally, an 8-DOF strongly nonlinear system under purely additive Poisson white noises is given as an example to highlight the accuracy of the proposed method. Results from the proposed method show good agreement with those from Monte Carlo simulations, which indicates that the proposed method can be applied to assess the response of high-DOF quasi-non-integrable Hamiltonian systems under Poisson white noises. In addition, if the potential energy of an arbitrary finite n-DOF system is the same form as that in Appendix B, pertinent high-dimensional domain integrals are converted into explicit expressions when n is even.