An integrated electric−gas system (IEGS) is the manifestation and development direction of a modern smart power system. This paper employs the cloud-edge computing method to research IEGS’s optimal dispatch to satisfy data protection requirements between power systems and natural gas systems and reduce data transmission pressure. Based on cloud-edge computing architecture, this paper constructs a cloud-edge computing method based on the Multi-agent Deep Deterministic Policy Gradient (MADDPG) algorithm to solve optimal dispatch problems. Then, this paper proposes an IEGS dispatch strategy based on cloud-edge computing, which conducts distributed computing independently at the edge of power and natural gas, and the cloud implements global dispatch based on boundary information and edge learning parameters. This method does not require the exchange of all information between the power system and natural gas system, effectively protecting data privacy. This paper takes the improved IEGS of the IEEE 9 node and Gas 8 node as an example to analyze. The equipment output of this dispatch method is within a reasonable range, and the cost is reduced by 0.21% to 1.03% compared with other methods, which verifies the effectiveness of the cloud-edge computing method in solving dispatch problems.