In 2020, China put forward the goals of “peak carbon dioxide emissions” and “carbon neutrality” (“double-carbon”) and it is urgent for the energy industry to achieve green transformation. Aiming at the rigid requirements of the carbon-peaking and carbon-neutrality goals (“double-carbon”), this study established a performance evaluation index system for an energy supply chain of a four-tier structure based on the “double-carbon” goals, calculating its weight by the analytic hierarchy process (AHP). On this basis, a three-stage data envelopment analysis (DEA) evaluation model was established to evaluate the performance of the energy supply chain in 2010–2019. According to the three-stage DEA evaluation mode, the initial input–output efficiency value of the energy supply chain was calculated by the DEA-BCC (extended by Banker, Charnes and Cooper) model and DEA-CCR (proposed by Charnes, Cooper and Rhodes) model and the influence of environmental noise was eliminated by stochastic frontier analysis (SFA) regression; we then obtained the adjusted efficiency value for the energy supply chain. At the same time, taking 2015 as the dividing point, the advantages and disadvantages between the traditional energy supply chain and new energy supply chain were analyzed and summarized. Further analysis and suggestions are provided to consumers, enterprises and countries from four aspects: energy supply, energy production and processing, energy transmission and distribution and energy consumption.