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As an important form of renewable energy utilization, microgrid (MG) is considered to be the main bearing form of distributed generation in the future. One of the most concerning issues in the operation of MG is how to realize its economic dispatch (ED). Nowadays, distributed algorithms have been increasingly used to solve the economic dispatch problem of MG. However, the MG based on the distributed optimization architecture must bear higher cyber‐attack risks. To address this issue, this paper investigates the distributed robust ED problem of MG. Firstly, a multi‐objective dispatch model of MG using a linear weighted sum (LWS) algorithm is developed, which considers the environmental and economic costs. On this basis, an event‐triggered fully distributed algorithm is proposed, which can effectively reduce communication times. Furthermore, an attack resilient strategy against false data injection (FDI) attacks is implemented in the proposed fully distributed algorithm, which has strong robustness against various colluding attacks and non‐colluding attacks, and can eliminate incorrect measurement of incremental cost and power generation data. Finally, the effectiveness of the proposed distributed control strategy is demonstrated through case studies in this paper. [ABSTRACT FROM AUTHOR]