Addressing carbon reduction in the energy sector is crucial in the global fight against climate change. In response to this, a source-load coordinated optimization framework is proposed for distributed energy systems (DES). The high carbon-emitting power plants in the source side are transformed into carbon capture power plants to capture CO2 generated during power generation, thereby improving the power efficiency and decreasing the carbon emissions of the DES. On the load side, the low carbon demand response (LCDR) method is introduced to replace the traditional price-driven demand response. Governed by dynamic carbon emission factors, LCDR aims to facilitate a low carbon shift in end users' energy consumption patterns. An extensive analysis is conducted on the viability of the proposed source-load coordinated framework for low-carbon economic scheduling and an optimal optimization model is formulated by considering the comprehensive cost of the DES. The original problem is then transformed into a hierarchical Stackelberg game model with multi-leaders and multi-followers, which is further solved by an efficient quasi-potential game (QPG) algorithm. The practicality and scalability of the proposed work are validated through simulations conducted on the modified IEEE39-node and IEEE118-node test systems. The findings verify that the proposed framework is highly effective in improving power plant efficiency, optimizing the use of renewable energy, and substantially lowering carbon emissions.