Service networks consisting of components and links are regarded as significant infrastructures to provide all kinds of services for users. They are supposed to select the best route for service requests in case of time-varying demand and different service patterns. In addition, it is necessary to choose appropriate routing metrics so as to satisfy user requirements for a variety of services. Multiple quality of service (QoS) requirements must be considered in the dynamic environment as a result of high demand for service networks. To that end, we propose an agent-based algorithm to address the routing problem in service networks. We construct a multi-layer network model to figure out component behaviors and complicated relationship between components under uncertainty. In order to reflect various service requirements, QoS metrics are defined from the perspectives of component and link. We also put forward an improved deep Q-learning method to achieve global convergence and enhance the efficiency of the routing algorithm. The numerical results on a case study illustrate the proposed algorithm finds high-quality solutions at acceptable costs, which routes service requests properly in the dynamic network environment. The proposed algorithm achieves outstanding performance compared with state-of-the-art routing algorithms in terms of delay and service factor.