This paper adopts an optimization-oriented exponential-polynomial-closure (OEPC) approach for conducting probabilistic analyses of oscillators under correlated multi-power velocity multiplicative excitation and additive excitation using the idea of exponential polynomial and for extending the conventional EPC method. While the EPC method uses the projection of the residue of the reduced FPK equation to formulate algebraic equations, the OEPC approach minimizes the residue square to handle the stochastic problem. The optimization process entails constructing an objective function (OBJ) with a weight function. The unknown coefficients in the exponential polynomial for estimating the probabilistic solution are then determined by minimizing the OBJ using a gradient-based method. The OEPC approach leads to an approximate probabilistic solution to the system response, which allows for statistical evaluations, such as the mean up-crossing rate. Four examples are provided to demonstrate the effectiveness of the OEPC approach in computing the asymmetric probabilistic solution of the stochastic oscillators with both odd and even nonlinear terms and subjected to correlated multi-power velocity multiplicative excitation and additive excitation. In particular, the verification of the solutions is done by comparison with both Gaussian closure method and Monte Carlo simulation to test the accuracy and reliability of the OEPC approach.