A novel method named optimization-oriented exponential-polynomial-closure approach is proposed in this article. The main idea of this attempt is to extend the original exponentialpolynomial-closure solution procedure methodologically by minimizing the resulted residual error square of the governing equation, which is achieved after an exponential polynomial is adopted as the approximate solution. The objective function for computing the parameters in the approximate solutions of nonlinear random oscillators is then formulated. The probabilistic solutions of the oscillators obtained by the presented approach are verified by the exact solutions in some special cases or by Monte Carlo simulation. Numerical examples indicate that the solutions attained by the presented approach match with the exact or Monte Carlo simulation solutions. The advantage of the presented solution procedure is that it can provide a much more accurate solution than the Equivalent Linearization approach and it is much more efficient than Monte Carlo simulation as demonstrated by the numerical examples.