This paper investigates the issues of the distributed energy management problem (EMP) of Microgrids in the presence of communication delays. To address this issue, a consensus-based distributed algorithm is proposed to integrate the economic dispatch and demand response, which can optimally assign the energy among generation units and load units with the objective of maximizing the total social welfare of the power system. Different from existing energy management algorithms, a nonuniform time-varying delays model is considered and embedded into the design of our algorithm, such that each unit can achieve collaborative optimization without the requirement of fixed delays information, which has both theoretical merits and practical engineering value for the efficient and stable operation of Microgrids. Moreover, it is proved that the proposed algorithm can converge to the optimal solution under some sufficient conditions. Finally, the correctness and effectiveness of the proposed method are validated by several simulation results.