Device-to-device (D2D) communication supports direct communications between nearby devices, which has potential to improve network capacity, spectrum efficiency and energy efficiency. Considering the high overhead to obtain the complete channel state information (CSI), we investigate resource allocation problems of D2D communication underlaying cellular networks with incomplete CSI to minimize the total power consumption. To deal with the challenge brought by incomplete CSI in estimating the instantaneous rates of D2D pairs (DPs), we consider two QoS metrics in terms of the expected rate and outage probability using statistical CSI. Based on that, we first investigate the energy-efficient power control problem for single cellular user (CU) and single DP sharing the same spectrum. With rigorous theoretical analysis of the intrinsic properties of CU rate and two QoS metrics, we design an optimal energy-efficient power control (EPO) algorithm for single CU and single DP. Using EPO as a building block, we then propose an energy-efficient resource allocation algorithm for multiple CUs and multiple DPs with incomplete CSI. Simulation results show that our algorithms consume the lowest powers compared with two baseline algorithms.