Previous studies have shown that the topological properties of a complex network, such as heterogeneity and average degree, affect the evolutionary game dynamics on it. However, traditional numerical simulations are usually time-consuming and demand a lot of computational resources. In this paper, we propose the method of dynamical approximate master equations (DAMEs) to accurately approximate the evolutionary outcomes on complex networks. We demonstrate that the accuracy of DAMEs supersedes previous standard pairwise approximation methods, and DAMEs require far fewer computational resources than traditional numerical simulations. We use prisoner's dilemma and snowdrift game on regular and scale-free networks to demonstrate the applicability of DAMEs. Overall, our method facilitates the investigation of evolutionary dynamics on a broad range of complex networks, and provides new insights into the puzzle of cooperation.