We consider the economic dispatch (ED) for an Energy Internet composed of energy routers (ERs), interconnected microgrids, and main grid. The microgrid consists of several bus nodes associated with distributed generators (DGs) and intelligent control units (ICUs). We propose a distributed ED algorithm for the grid-connected microgrid, where each ICU iterates the estimated electricity price of the distribution system and the estimation for the average power mismatch of the whole microgrid by leader-following and average-consensus algorithms, respectively. The ER iterates the incremental power exchanged with the distribution system. By constructing an auxiliary consensus system, we prove that if the communication topology of the Energy Internet contains a spanning tree with the ER as the root and there is a path from each ICU to the ER, then the estimated electricity price of the distribution system converges to its real value, the power supply and demand achieves balance and the ED achieves optimal asymptotically. Furthermore, we propose an autonomous distributed ED algorithm covering both grid-connected and isolated modes of the microgrid by feeding back the estimated average power mismatch for updating the incremental costs with penalty factor. It is proved that if the communication topology of the microgrid is connected and there exists an ICU bidirectionally neighboring the ER, then the microgrid can switch between the two modes reliably. The simulation results demonstrate the effectiveness of the proposed algorithms.