To mitigate the global warming threat, CO2 emission reduction is an irreversible trend for the sustainable development of power systems. Among various low-carbon technologies, gas-fired power plants and power-to-gas facilities play an important role to reduce emissions, and they are increasing the interdependency between electricity and natural gas systems. Considering also the increasing penetration of wind power generation, this paper proposes a low-carbon economic dispatch model under both constraints of the electricity and natural gas systems. To reduce CO2 emission and improve the wind power utilization, mathematical formulations of the post-combustion carbon capture system and power-to-gas facility are presented in the proposed model. Additionally, a flexible operation mode of post-combustion carbon capture system and power-to-gas facility is further analyzed. The objective function of the presented model is to minimize the total cost, which consists of the operation cost, the CO2 processing cost and the penalty cost of wind power curtailment. Then the optimization model is converted into a mixed integer linear programming problem for efficient computation purpose. Numerical case studies are carried out to validate the effectiveness of the proposed model and the flexible operation mode.