At present, online shopping is becoming increasingly popular. In particular, low-carbon products are becoming more favored as consumers' low-carbon awareness increases. Manufacturers sell their low-carbon products through e-commerce platforms. Thus, the manufacturer and the e-commerce platform form a low-carbon e-supply chain system. The manufacturer makes products with carbon emission reduction efforts, while the e-commerce platform provides a sales service. In this paper, we described models for a decentralized decision mode and a centralized decision mode in the low-carbon e-supply chain, and compare the decision results. Our findings show that the centralized decision mode has a better performance than the decentralized one, the ability of the e-supply chain to respond to consumers' preference for a low-carbon product has a direct effect on its operation, and the manufacturer's carbon emission reduction behavior will be a potential source for enterprises to gain more revenue. Further, we designed a coordination contract for them that can be accepted by both sides in the decentralized decision mode. We find that if the e-commerce platform can share the carbon emission reduction costs of the manufacturer, the performance of the e-supply chain will be greatly improved. A practical case study and numerical examples validate our analysis.