In this work, we propose a new deep imitation learning (DIL)-driven edge-cloud computation offloading framework for MEC networks. A key objective for the framework is to minimize the offloading cost in time-varying network environments through optimal behavioral cloning. Specifically, we first introduce our computation offloading model for MEC in detail. Then we make fine-grained offloading decisions for a mobile device, and the problem is formulated as a multi-label classification problem, with local execution cost and remote network resource usage consideration. To minimize the offloading cost, we train our decision making engine by leveraging the deep imitation learning method, and further evaluate its performance through an extensive numerical study. Simulation results show that our proposal outperforms other benchmark policies in offloading accuracy and offloading cost reduction. At last, we discuss the directions and advantages of applying deep learning methods to multiple MEC research areas, including edge data analytics, dynamic resource allocation, security, and privacy, respectively.