With the rising environmental concerns among consumers all over the world, sustainability has received considerable attention, and numerous enterprises are adopting various practices such as investing in energy-saving to improve sustainability in supply chains. However, many previous researches always assume that decision makers are perfectly rational and neglect the behavioral concerns of decision makers. This paper considers a two-stage sustainable supply chain with behavioral concerns in order to develop more realistic models, and mainly focuses on the energy-saving and pricing decisions in the decentralized system, as well as how to improve energy-saving level and profits. We develop decentralized decision-making models under two types of behavioral concerns: fairness concern and risk aversion, and derive the optimal strategy for each member with a Stackelberg game in which the manufacturer acts as the leader. The effect of the behavioral concerns on the optimal decisions and corresponding profits is discussed in detail. Theoretical analysis verified by numerical experiments shows that the fairness behavior always causes a negative effect on the manufacturer, total supply chain, and energy conservation, while it could benefit the retailer in profits. The risk aversion behavior always benefits the manufacturer, total supply chain, and energy conservation, whereas it could make the retailer suffer. Note that both the optimal energy-saving level and corresponding profit of the total supply chain under two types of behavioral concerns are lower than that in the centralized system, thereby we propose a revenue-cost-sharing contract to coordinate the supply chain, under which both the manufacturer and the retailer can achieve a win-win outcome and the energy-saving level can be improved. In addition, some managerial implications through our analytical and numerical results are summarized in this paper.