A novel resource allocation (RA) algorithm for non-orthogonal transmission is proposed to maximize overall throughput of user equipments (UEs) in unmanned aerial vehicle (UAV) relay systems (URSs), which jointly optimizes transmit duration and power of signals at base station and UAV. Also, the proposed RA algorithm is shown to maximize the overall throughput of UEs while guaranteeing a seamless UAV relay transmission through the analysis of outage probability and power control strategy. Simulation results reveal that the proposed RA algorithm for non-orthogonal transmission in URSs achieves the maximum throughput of UEs within the actual operation range of UAV.