It is well-known that power networks are prone to cascading failures in the event of a network disruption. A traditional way to prevent such cascading failures is to shed some of the supportable demand from the network. Interestingly, due to the non-local nature of power flow distributions, further deliberate disconnection of lines in a disrupted power network can result in improvements in its supportable demand, the phenomenon resembling Braess' paradox. In this paper, we exploit this phenomenon to formulate a multi-stage control scheme using: a fast timescale, linear Network Stabilization Problem based on demand shedding; and a slow timescale, combinatorial Demand Maximization Problem based on link shedding. We provide a key example illustrating the paradox and some structural results demonstrating the limitations and potential of our control scheme. Using Simulated Annealing to tackle the large combinatorial problem, we also investigate the efficacy of our approach for the Polish power grid (which consists of 2383 buses and 2896 lines).