A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.