If an end-to-end (E2E) path includes multiple domains, we need inter-domain collaboration to ensure the E2E quality-of-service (QoS) for the applications. In heterogeneous networks, an E2E path may go through domains with several QoS classes in each domain. However, the prevalent legacy network architecture and the standard software-defined networking (SDN) model lack effective mechanisms for inter-domain collaboration and QoS class mapping. In this study, we propose a hierarchical SDN control plane approach to guarantee the E2E QoS among multiple domains with various QoS classes on the E2E path. We propose a controller module for selecting the most suitable QoS class for each domain in the E2E path based on multi-criteria decision-making by using the technique for order of preference by similarity to ideal solution (TOPSIS). We map the suitable service classes in the global controller (GC) for provisioning the E2E QoS according to the application service requests. First, we propose an SDN-based inter-domain communication scheme and the message processing algorithm for E2E service delivery when multiple QoS classes exist in each domain. Next, we formulate the problem of service class selection with TOPSIS, provide an E2E mapping scheme, and demonstrate it with an example. Finally, we compare the proposed approach with the existing schemes for E2E QoS class mapping in terms of E2E delay, jitter, packet loss rate (PLR), and cost (per bandwidth unit). According to our simulation results, the proposed approach ensures the E2E QoS and guarantees the E2E delay, jitter, PLR, and cost according to the application service requests.