This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.