Edge computing technology can effectively solve huge challenges posed by the large number of terminal devices accessing and massive data processing in digital distribution networks. Burst tasks, such as faults and data requests from the cloud, can occur at any time for edge computing devices in distribution networks. These tasks are unpredictable and usually hold the highest priority and must be completed as soon as possible. Although resources can be reserved partially at each period in the pre-scheduled operation plan, they may still be insufficient to handle burst tasks adequately. A real-time resource dispatch approach for burst tasks is developed in this study to address the above problems. The concept of flexibility for edge computing devices is presented, determining the real-time dispatch duration. Real-time resource dispatch and task handling processing are analyzed in detail, considered as task real-time dispatch models, computation process real-time dispatch constraints, and resource limitation constraints. The proposed real-time resource dispatch approach takes full advantage of the transferable characteristics for partial original plan tasks to adjust the pre-scheduled operation plan and release flexible resources for immediate processing of the burst task, completing burst tasks quickly and minimizing the impact for previous planned tasks on the edge computing device. The capability of the proposed method to efficiently deal with the burst tasks is also verified by the case study.