The distributed economic dispatch of microgrids is a core problem in smart grids, which aims to minimize the total generation cost while satisfying power balance and operational constraints of generation units. Compared with centralized methods that rely on a central controller to collect global information from all distributed agents, the distributed economic dispatch scheme offers advantages including strong scalability, low communication overhead, and enhanced privacy preservation. To address the slow convergence rate of existing distributed economic dispatch algorithms, this paper proposes a distributed Nesterov-accelerated economic dispatch algorithm via integrating the Nesterov acceleration technique with a primal-dual framework. Specifically, the adaptive dual variable is introduced to handle local output constraints, and the dynamics of primal and dual variables are designed via the Nesterov acceleration technique. Therefore, theoretical analysis shows that the proposed algorithm can achieve the global optimal solution with a convergence rate of $O(\frac{1}{t^2})$ under generally convex cost functions. Finally, simulation results demonstrate that the proposed algorithm outperforms existing distributed Nesterov economic dispatch algorithms.