The large network transmission delay and high energy consumption in cloud computing as well as the limited computing resource in edge servers are the bottlenecks for the development of delay-sensitive Internet of Things (IoT) applications. In order to improve the Quality of Service (QoS) of IoT applications while achieving green computing for computing systems, an edge-cloud cooperation Drift-plus-Penalty-based Computation Offloading (DPCO) policy was proposed. Firstly, mathematical modeling was performed on the business model, the transmission delay as well as the computation delay of the computation job, the computation energy as well as the transmission energy generated by the system were modeled by constructing the IoT-Edge-Cloud model. Then, the system consumption and the job average delay were optimized, with the queueing stability of the edge servers as constraint condition, the edge-cloud cooperation computation offloading optimization model was built. After that, with the optimization targets as the penalty function, the drift-plus-penalty function properties of computation offloading optimization model were analyzed based on Liapunov stability theory. Finally, DPCO was proposed based on the above results, the long-term energy consumption per unit time and the average system delay were reduced by selecting the computation offloading policy of minimizing the present drift-plus-penalty function in every time slot. In comparison with Light Fog Processing (LFP), the benchmarked Edge Computing (EC) and Cloud Computing (CC) policies, DPCO consumes the lowest system energy, which is 2/3 of that of the CC policy; DPCO also provides the shortest average job delay, which is 1/5 of that of the CC policy. The experimental results show that DPCO can efficiently reduce the energy consumption of edge-cloud computing system, shorten the end-to-end delay of the computation job, and satisfy the QoS requirements of delay-sensitive IoT applications.