Emotions are indispensable in human communication, but are often overlooked in task-oriented dialogue (ToD) modelling, where the task success is the primary focus. While existing works have explored user emotions or similar concepts in some ToD tasks, none has so far included emotion modelling into a fully-fledged ToD system nor conducted interaction with human or simulated users. In this work, we incorporate emotion into the complete ToD processing loop, involving understanding, management, and generation. To this end, we extend the EmoWOZ dataset (Feng et al., 2022) with system affective behaviour labels. Through interactive experimentation involving both simulated and human users, we demonstrate that our proposed framework significantly enhances the user's emotional experience as well as the task success.