Many computer network applications impose constraints for multiple quality of service (QoS) metrics, e.g., delay, packet loss, bandwidth, and jitter. These QoS constraints cannot be guaranteed by the Internet due to its best-effort service model. Overlay networks have been an effective technique at the application layer to support multiple QoS constraints of networking applications. In software-defined overlay networks, software-defined networking (SDN) paradigm is introduced in the overlay networks to enable centralized and efficient routing of traffic in the overlay networks, thus, enabling better QoS. One of the main challenges in software-defined overlay networks is the fast-changing overlay link QoS characteristics. However, the existing routing algorithms for satisfying multiple QoS constraints in software-defined overlay networks involve high route computation time and thus these routing algorithms cannot adapt to the fast-changing overlay link QoS characteristics. Moreover, as we scale the size of overlay networks, the size of forwarding tables increases exponentially. This is because the existing routing schemes for ensuring multiple QoS constraints use both the source and the destination address for data-plane forwarding. This leads to pushing a huge amount of forwarding table entries by the controller through the network and thus limiting the size of the overlay network. We propose an efficient routing scheme, QROUTE, for satisfying multiple QoS constraints in software-defined overlay networks. QROUTE consists of a control plane routing algorithm which has significantly low route computation time because of employing a novel directed-acyclic-graph (DAG) based approach. QROUTE also reduces the forwarding entries in the data plane by using a QoSmetric-based forwarding scheme. We extensively evaluate QROUTE using traces from a global overlay service provider. We also examine QROUTE on a testbed of P4-BMv2 switches controlled by the ONOS controller using P4Runtime protocol. We find that QROUTE outperforms other state-of-the-art QoS routing schemes in route computation time, size of the forwarding tables and meeting the QoS requirements of various applications.