Existing dialogue corpora and models are typically designed under two disjoint motives: while task-oriented systems focus on achieving functional goals (e.g., booking hotels), open-domain chatbots aim at making socially engaging conversations. In this work, we propose to integrate both types of systems by Adding Chit-Chat to ENhance Task-ORiented dialogues (ACCENTOR), with the goal of making virtual assistant conversations more engaging and interactive. Specifically, we propose a Human <-> AI collaborative data collection approach for generating diverse chit-chat responses to augment task-oriented dialogues with minimal annotation effort. We then present our new chit-chat-based annotations to 23.8K dialogues from two popular task-oriented datasets (Schema-Guided Dialogue and MultiWOZ 2.1) and demonstrate their advantage over the originals via human evaluation. Lastly, we propose three new models for adding chit-chat to task-oriented dialogues, explicitly trained to predict user goals and to generate contextually relevant chit-chat responses. Automatic and human evaluations show that, compared with the state-of-the-art task-oriented baseline, our models can code-switch between task and chit-chat to be more engaging, interesting, knowledgeable, and humanlike, while maintaining competitive task performance.