To combat global warming caused by excessive CO2 emissions, it is critical to achieving low-carbon economic operation of integrated electricity and heat systems (IEHS). Thermal power plants with carbon capture systems can significantly reduce the spread of carbon emissions, thus effectively reducing the overall carbon footprint. Meanwhile, the development and utilization of renewable energy sources, such as wind power, can effectively limit overall carbon emissions. Hence, this paper proposes a multi-objective optimization scheme incorporating carbon capture power plants (CCPP) and multi-energy demand response (MEDR), considering the uncertainties of the system for the economic-emission dispatch problem in IEHS. First, the energy flow mechanism and analysis of the wind power abandonment of the IEHS are presented. Then, the low-carbon mechanisms of the MEDR and CCPP are investigated. The uncertainties of the system are described by a membership function to ensure the reserve requirements. A weight setting method based on the improved analytic hierarchy process method is proposed, in which the multi-objective function is transformed into a single-objective function. The case studies were tested on an improved IEEE-30 bus power system and a 6-bus district heating system. The results demonstrate the effectiveness of the proposed method in reducing carbon emissions and improving wind power consumption.