This paper deals with the stationary probability density function solution of the dynamic response for nonlinear oscillators excited by Poisson impulses. The probability density function solution is governed by the generalized Fokker–Planck–Kolmogorov equation. The exponential-polynomial closure method is used to obtain an approximate solution to the generalized Fokker–Planck–Kolmogorov equation. To evaluate the effectiveness of the exponential-polynomial closure method in the case of Poisson excitations, Duffing oscillators, van der Pol oscillators, and an additional nonlinear oscillator excited by external and parametric Poisson impulses with different levels of nonlinearity in stiffness and damping are studied. The numerical results show good agreement with the probability distribution obtained with Monte Carlo simulations including the tail regions, which is of significance for reliability analyses.