Weighted vertex cover (WVC), a generalized type of vertex cover, is one of the most important combinatorial optimization problems. In this paper, we provide a novel solution to the WVC problem from the view of network engineering. We model the WVC problem as an asymmetric game on weighted networks, where each vertex is treated as an intelligent rational agent rather than an inanimate one. Under the framework of asymmetric game, we find that strict Nash equilibriums of the asymmetric game are the intermediate states between the WVC states and the minimum WVC (MWVC) states. Besides, we propose best response algorithms with memory and feedback to solve the WVC problem, and find that a better approximate solution to the MWVC can be obtained under the feedback-based best response algorithm. Numerical illustrations verify the performance of the proposed game solution on weighted networks. Our findings pave a new way to solve the WVC problem from the perspective of asymmetric game, which opens a bottom-up avenue to address the combinatorial optimization problems.