This study constructs a two-echelon low-carbon supply chain consisting of a single manufacturer and a retailer. The manufacturer’s initial capital is zero. The study analyzes three financing modes of the manufacturer by using the Stackelberg game model: external financing (the EF mode), internal financing (the IF mode), and trade credit financing (the TC mode). We found that the IF mode is superior to the EF mode while inferior to the TC mode regarding carbon emission reduction (CER), market demand, and the manufacturer’s profit. Additionally, the IF mode is superior to the EF mode regarding the retailer and supply chain’s profits and special to the TC mode only when the loan and deposit interest rate meet the specific conditions. The best financing mode for the manufacturer is the TC mode, but he needs to give some compensation to the retailer in order to make the retailer cooperate with him. The impact of carbon tax rate changes on the equilibrium solutions in the three financing modes is not only related to the change range of the carbon tax rate itself but also closely related to the initial carbon emissions. A high carbon tax rate can help stimulate the cleaner manufacturer to reduce carbon emissions, but a moderate carbon tax rate is more applicable to the polluting manufacturer. A numerical example is given to demonstrate some of the conclusions. Finally, the study provides managerial insights based on the analytical results.