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Reconfigurable hardware, such as Field Programmable Gate Arrays (FPGAs), are being increasingly deployed in diverse application areas including automotive systems, critical infrastructures, and the emerging Internet of Things (IoT), to implement customized designs. However, securing FPGA-based designs against piracy, reverse engineering, and tampering is challenging, especially for systems that require remote upgrade. In many cases, existing solutions based on bit-stream encryption may not provide sufficient protection against these attacks. In this paper, we present a novel obfuscation approach for provably robust protection of FPGA bitstreams at low overhead that goes well beyond the protection offered by bitstream encryption. The approach works with existing FPGA architectures and synthesis flows, and can be used with encryption techniques, or by itself for power and area-constrained systems. It leverages “FPGA dark silicon” - unused resources within the configurable logic blocks - to efficiently obfuscate the true functionality. We provide a detailed threat model and security analysis for the approach. We have developed a complete application mapping framework that integrates with the Altera Quartus II software. Using this CAD framework, we achieve provably strong security against all major attacks on FPGA bitstreams with an average 13% latency and 2% total power overhead for a set of benchmark circuits, as well as several large-scale open-source IP blocks on commercial FPGA.