In this paper, we asses the benefits of using partial relaying in energy-harvesting networks. We consider a system composed of a source, a relay, and a destination. Each of the source and the relay has energy-harvesting capability and generates its own traffic. The source is helped by the relay through a partial relaying network-level cooperation protocol. The relay regulates the arrivals from the source by accepting only a proportion of the successfully received packets at the relay. The relaying parameter, which determines the proportion of packets to be accepted, is selected based on the parameters of the network to ensure the stability of the source and the relay data queues. In this work, we provide an exact characterization of the stability region of the network. We derive the optimal value of the relaying parameter to maximize the stable throughput of the source for a given data arrival rate to the relay. Also, we compare the stability region of the proposed strategy with partial relaying to the stability regions of simple transmission strategies. Finally, we consider the problem of network utility optimization in which we optimize over the value of the relaying parameter for a given pair of data arrival rates for the source and the relay.