In view of power outages caused by extreme events such as natural disasters and cyber attacks, and a reduction of the load loss rate and economic losses of the power grid, an active distribution network power supply restoration strategy has thus been proposed with energy storage integrated soft open point (E-SOP) combined with network reconstruction under extreme events. Firstly, with an introduction of the topology diagram of E-SOP connected to the distribution network, its mathematical model under fault recovery is established. Secondly, taking into consideration the importance of the tertiary load and network losses, with the objective function of minimizing the loss of each node in the distribution network and minimizing network losses, a joint operation model of E-SOP and network reconstruction for power supply recovery is constructed for a rapid recovery of the active distribution network under extreme events. Finally, taking the improved IEEE33 node distribution network as an example, a research is carried out on the effects of E-SOP installation parameters such as capacity, location, and quantity on the restoration capability, followed by a validation and analysis of the proposed power restoration strategy. The research results indicate that the proposed strategy helps to reduce the loss of load rate under distribution network faults, thus further improving the level of power supply recovery.