The promotion of energy conservation and emission reduction involves a multi-party game among governments, enterprises, and other stakeholders. To explore the game relationships among governments, the public, and coal-fired power enterprises under the “dual carbon targets”, this paper constructs an evolutionary game model for energy conservation and emission reduction involving three parties: the government, coal-fired power enterprises, and the public. Through a theoretical analysis and simulation analysis of the case study involving a central Hebei energy enterprise in China, the impact of parameter variations on the strategic choices of all parties and the evolutionarily stable strategies of the system is thoroughly discussed. The research findings indicate that reducing public supervision costs, increasing government rewards, subsidies, and penalties, and enhancing government regulatory capabilities are crucial factors in promoting energy-saving and emission-reduction efforts by coal-fired power enterprises. After multiple evolutionary iterations, the tripartite evolutionary game system ultimately reaches an evolutionarily stable state of government regulation, public supervision, and energy-saving and emission-reduction by coal-fired power enterprises at the point E8(1,1,1). Based on these findings, we propose a series of policy recommendations aimed at providing theoretical support for the Chinese government to achieve its energy-saving and emission-reduction strategies under the dual-carbon targets. These recommendations also offer practical guidance for the government in formulating emission reduction policies, for enterprises in optimizing their operational strategies, and for the public in participating in emission reduction efforts.