As awareness of environmental protection increases worldwide, enterprises have been building their supply chains in ways that conserve natural resources and minimize the creation of pollutants. One of the practical ways to make supply chains more sustainable is for enterprises to utilize green innovation strategies and to increase resource reuse. In this work, we focus on a closed-loop supply chain (CLSC) consisting of a manufacturer, a retailer, and a collector. In the investigated CLSC, the manufacturer and the retailer drive the green innovation strategy either individually or simultaneously to boost market demand. In the reverse flow of the CLSC, the collector is responsible for collecting consumers’ used products and transferring them to the manufacturer for remanufacturing. By combining two types of the market leadership and three types of green innovation strategies, we establish six different Stackelberg game models and solve them analytically. Through an extensive comparative analysis, we show who should have market leadership and who should drive the green innovation strategy in the CLSC. Various numerical examples are also given to support our major findings. One of our key findings suggests that the supply chain members must participate in green innovation activities at the same time to achieve a win-win scenario in the CLSC.