The green management of megaprojects can mitigate the impact of construction activities on the regional ecological environment. However, given the long construction timelines and the difficulty in accurately predicting potential environmental damage, key stakeholders (i.e., contractors, owners, and environmental agencies) involved in the green management of the construction phase are susceptible to uncertainties when making green management decisions. This study integrates stochastic processes into evolutionary game theory, reconstructing a stochastic evolutionary game model among contractors, owners, and environmental agencies, exploring the strategic decision-making processes of these stakeholders in green management and analyzing the effects of various factors on their strategic choices. The findings reveal that if contractors adopt green construction, owners pursue service-oriented strategies, and environmental agencies conduct random inspections, a stable strategy set for green management would be achieved in megaprojects. Especially, strengthening the initial willingness of all players to cooperate in green management would effectively reduce decision-making volatility. Additionally, lower total environmental protection costs and a more reasonably proportion of these costs borne by owners could accelerate the achievement of a stable state in green management. Enhancing the social reputation of contractors could incentivize them to adopt green construction strategies, while positive feedback from contractors to owners could foster the development of service-oriented ownership. Moreover, the green management decisions of contractors and owners are particularly sensitive to random disturbances. The stochastic evolutionary game model proposed in this study extends traditional approaches to green management, and the vision of service-oriented ownership provides valuable insights into establishing long-term green management mechanisms in the practices of megaprojects.