A national strategy has been deployed for the development of China’s western region. The Chengdu–Chongqing city cluster is an important platform for collaborative regional innovation. In this empirical study, we constructed multilevel innovation networks for each of the 16 cities in the Chengdu–Chongqing city cluster and the inter-city networks for these 16 cities based on a panel of data on applications for invention patents by industry–university–research collaborators in the city cluster. We used social network analysis and a negative binomial regression model with fixed effects to examine the impact of the multilevel innovation networks on the innovation performance of the 16 cities. The moderating effect of government support was also analyzed. The results show that the average weighted degree of the intra-city innovation network has significant positive effects on the innovation performance of the 16 cities. For the inter-city innovation networks, the network density and cooperation intensity have significant positive effects on the innovation performance of the 16 cities. Regarding the moderating effect, our results show that a high level of government support enhances the positive effects of the average weighted degree of the intra-city innovation networks and the network density of the inter-city innovation networks on the innovation performance.