With the assistance of edge-cloud computing, a pre-assigned task can be migrated among edge-cloud servers to improve the efficiency of task processing. Since traditional heuristic optimization methods usually take multiple iterations, they cannot adapt to the dynamic environment. Recently, deep reinforcement learning (DRL) has gained significant attention to solve the task decision making problem. The neural network frameworks can extract potential features from input task data in the edge-cloud computing. However, the cloud-edge network’s structure information features are ignored. To tackle this issue, we propose an efficient task pre-assignment and migration (ETPAM) algorithm based on DRL, which takes the advantage of graph-based relational inference capability from graph convolutional networks (GCN) and the self-evolution capability of Soft Actor-Critic (SAC). ETPAM could choose the edge/cloud servers adaptively to migrate the pre-assigned task, by interacting with the edge-cloud environment. Simulation results demonstrate that, compared with three DRL based methods, our ETPAM can effectively and efficiently generate near-optimal migration decisions with lower energy consumption, response time and service-level agreement (SLA) violation rate.