Computation offloading is generally regarded as a promising technology to address the problem of insufficient computing power in mobile devices, while simultaneously satisfying low-latency requirements and furnishing exceptional support for intelligent applications. However, with the advancement of computing-intensive and delay-sensitive application service demands, how to assign applications with diversity requirements among various edge servers still remains a challenge. In this paper, we propose a graph neural network (GNN)-based dependency-aware task scheduling and offloading (GNN-TSO) scheme for MEC-assisted network to effectively coordinate wireless and computing resources for multiple applications. We first model the dependencies among all tasks of an application as a directed acyclic graph (DAG) and formulate the dependency-aware task scheduling and offloading problem as a combinatorial optimization problem which is hard to be solved. To capture the scalable features of DAG-type applications, we introduce the GNN to process tasks information of applications. Then we construct a Markov decision process for the fine-grained task scheduling and offloading strategy and apply policy gradient algorithm to jointly optimize the task scheduling priority and computation offloading decision. Simulation results show that the proposed scheme can effectively reduce the network cost compared with other reference schemes.