Edge computing is a new paradigm, which provides storage, computing, and network resources between the traditional cloud data center and terminal devices. In this paper, we concentrate on the application-driven task offloading problem in edge computing by considering the strong dependencies of sub-tasks for multiple users. Our objective is to joint optimize the total delay and energy generated by applications, while guaranteeing the quality of services of users. First, we formulate the problem for the application-driven tasks in edge computing by jointly considering the delays and the energy consumption. Based on that, we propose a novel Application-driven Task Offloading Strategy (ATOS) based on deep reinforcement learning by adding a preliminary sorting mechanism to realize the joint optimization. Specifically, we analyze the characteristics of application-driven tasks and propose a heuristic algorithm by introducing a new factor to determine the processing order of parallelism sub-tasks. Finally, extensive experiments validate the effectiveness and reliability of the proposed algorithm. To be specific, compared with the baseline strategies, the total cost reduction by ATOS can be up to 64.5% on average.