Integrating carbon trading mechanisms with generalized energy storage (GES) fully embodies the principles of green and coordinated development, serving as a crucial means to achieve low-carbon construction of microgrids. This research presents a strategy for optimizing energy allocation within microgrids to minimize carbon emissions and enhance microgrid systems' economic-environmental benefits. The strategy takes into account the use of tiered carbon trading and GES. Based on a typical microgrid system architecture, an economic dispatch model for microgrids is developed, which integrates renewable energy sources such as wind and solar storage, gas turbines, energy storage systems, and flexible resources on the demand side. The model aims to minimize carbon emissions while optimizing the allocation of resources. Subsequently, the model facilitates microgrid carbon emission control by considering the transferable, convertible, and reducible properties of GES. Furthermore, implementing a tiered carbon trading mechanism decreases carbon emissions. Finally, using a real microgrid example from a specific region in China, the results indicate that the proposed method significantly enhances the system's low-carbon level. Notably, compared to scenarios that do not consider GES, the proposed method substantially reduces total costs by 6.62% and decreases carbon emissions by 22.2%. The findings indicate that the suggested dispatch model can substantially decrease carbon emissions while simultaneously improving the economic efficiency of the microgrid system.