In this paper, we consider a wireless-powered cooperative communication network consisting of one hybrid access-point (AP), one source, and one relay. In contrast to conventional cooperative networks, the source and relay in the considered network have no embedded energy supply. They need to rely on the energy harvested from the signals broadcasted by the AP for their cooperative information transmission. Based on this three-node reference model, we propose a harvest-then-cooperate (HTC) protocol, in which the source and relay harvest energy from the AP in the downlink and work cooperatively in the uplink for the source's information transmission. Considering a delay-limited transmission mode, the approximate closed-form expression for the average throughput of the proposed protocol is derived over Rayleigh fading channels. Subsequently, this analysis is extended to the multi-relay scenario, where the approximate throughput of the HTC protocol with two popular relay selection schemes is derived. The asymptotic analyses for the throughput performance of the considered schemes at high signal-to-noise radio are also provided. All theoretical results are validated by numerical simulations. The impacts of the system parameters, such as time allocation, relay number, and relay position, on the throughput performance are extensively investigated. ; Comment: Accepted by IEEE Trans. Signal Processing