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Mobile edge computing (MEC)-enabled satellite–terrestrial networks (STNs) can provide task computing services for Internet of Things (IoT) devices. However, since some applications’ tasks require huge amounts of computing resources, sometimes the computing resources of a local satellite’s MEC server are insufficient, but the computing resources of neighboring satellites’ MEC servers are redundant. Therefore, we investigated inter-satellite cooperation in MEC-enabled STNs. First, we designed a system model of the MEC-enabled STN architecture, where the local satellite and the neighboring satellites assist IoT devices in computing tasks through inter-satellite cooperation. The local satellite migrates some tasks to the neighboring satellites to utilize their idle resources. Next, the task completion delay minimization problem for all IoT devices is formulated and decomposed. Then, we propose an inter-satellite cooperative joint offloading decision and resource allocation optimization scheme, which consists of a task offloading decision algorithm based on the Grey Wolf Optimizer (GWO) algorithm and a computing resource allocation algorithm based on the Lagrange multiplier method. The optimal solution is obtained by continuous iterations. Finally, simulation results demonstrate that the proposed scheme achieves relatively better performance than other baseline schemes.