Device-to-device (D2D) communication enables simultaneous data transmissions by cellular users (CU) and D2D user pairs, but at the expense of additional interference between them. The literature on resource allocation in D2D systems often assumes that the base station (BS) has complete channel state information (CSI) about all the links between all the users in a cell. However, acquiring the CSI of cross links between the CUs and the D2D receivers is a critical bottleneck because the number of cross links is the product of the number of CUs and D2D pairs. We study a novel partial CSI model in which the overhead of feeding back the CSI of the cross links is much lower. For a cell with one D2D pair and multiple CUs, we propose a novel throughput-optimal joint mode selection, user scheduling, and rate adaptation policy that exploits information about the statistics of the cross links and incorporates inter-cell interference. We derive closed-form expressions for the feedback-conditioned goodput for the underlay mode, which drives this optimal policy. We also present extensions that incorporate user fairness, quantized CSI, and multiple D2D pairs and multiple subchannels.