To accommodate the explosive growth in mobile data traffic, both mobile cellular operators and mobile users are increasingly interested in offloading the traffic from cellular networks to Wi-Fi networks. However, previously proposed offloading schemes mainly focus on reducing the cellular data usage, without paying too much attention on the quality of service (QoS) requirements of the applications. In this paper, we study the Wi-Fi offloading problem with delay-tolerant applications under usage-based pricing. We aim to achieve a good tradeoff between the user's payment and its QoS characterized by the file transfer deadline. We first propose a general Delay- Aware Wi-Fi Offloading and Network Selection (DAWN) algorithm for a general single-user decision scenario. We then analytically establish the sufficient conditions, under which the optimal policy exhibits a threshold structure in terms of both the time and file size. As a result, we propose a monotone DAWN algorithm that approximately solves the general offloading problem, and has a much lower computational complexity comparing to the optimal algorithm. Simulation results show that both the general and monotone DAWN schemes achieve a high probability of completing file transfer under a stringent deadline, and require the lowest payment under a non-stringent deadline as compared with three heuristic schemes.