The actual situation of carbon-emission reduction in China’s power sector has not yet achieved the expected benefits. The rent-seeking behavior of participants in power construction projects (PCPs) hinders the realization of low-carbon benefits. It is necessary to explore the behavioral strategies of the various participants in the low-carbon transition of PCPs. This paper creatively constructs an evolutionary game model of PCPs’ participants from the perspective of MRV (monitoring, reporting, and verification) and introduces the influence of the public to provide a comprehensive analysis of strategic equilibrium points. Through numerical simulations with MATLAB R2021a software, this paper explores the strategic choices of participants in different situations and gives relevant inferences and proofs. The results show that the grid company dominates at the initial stage and promotes participants to regulate behaviors. Under the premise of satisfying the system-stability requirements, setting the growth rate of the grid company’s punishments to 100% can enhance the willingness for strict supervision, while the growth rate of the supervision costs to 200% significantly decreases the probability of strict supervision. With the integration of MRV and PCPs, participants spontaneously fulfill the carbon-emission-reduction tasks. Reasonable control of input costs can effectively avoid the occurrence of rent-seeking behavior. In addition, this paper sets the public-influence growth rate at 200% and finds that the public plays a greater role in driving participants to fulfill responsibilities. Based on the results, a low-carbon transition mechanism for PCPs under the MRV system is proposed by considering several dimensions, which provides suggestions for participants to fulfill carbon-reduction responsibilities.