Networks are powerful instruments to study complex phenomena, but they become hard to analyze in data that contain noise. Network backbones provide a tool to extract the latent structure from noisy networks by pruning non-salient edges. We describe a new approach to extract such backbones. We assume that edge weights are drawn from a binomial distribution, and estimate the error-variance in edge weights using a Bayesian framework. Our approach uses a more realistic null model for the edge weight creation process than prior work. In particular, it simultaneously considers the propensity of nodes to send and receive connections, whereas previous approaches only considered nodes as emitters of edges. We test our model with real world networks of different types (flows, stocks, cooccurrences, directed, undirected) and show that our Noise-Corrected approach returns backbones that outperform other approaches on a number of criteria. Our approach is scalable, able to deal with networks with millions of edges.