Fifth generation (5G) cellular networks will be comprised of millions of connected devices like wearable devices, Androids, iPhones, tablets, and the Internet of Things (IoT) with a plethora of applications generating requests to the network. The 5G cellular networks need to cope with such sky-rocketing traffic requests from these devices to avoid network congestion. As such, cloud radio access networks (C-RAN) has been considered as a paradigm shift for 5G in which requests from mobile devices are processed in the cloud with shared baseband processing. Despite call admission control (CAC) being one of radio resource management techniques to avoid the network congestion, it has recently been overlooked by the community. The CAC technique in 5G C-RAN has a direct impact on the quality of service (QoS) for individual connections and overall system efficiency. In this paper, a novel fuzzy logic-based CAC scheme with preemption in C-RAN is proposed. In this scheme, cloud bursting technique is proposed to be used during congestion, where some delay tolerant low-priority connections are preempted and outsourced to a public cloud with a penalty charge. Simulation results show that the proposed scheme has low blocking probability below 5%, high throughput, low energy consumption, and up to 95% of return on revenue.