Signaling networks often encounter multiple ligands and must respond selectively to generate appropriate, context-specific outcomes. At thermal equilibrium, ligand specificity is limited by the relative affinities of ligands for their receptors. Here, we present a non-equilibrium model showing how signaling networks can overcome thermodynamic constraints to preferentially signal from specific ligands while suppressing others. In our model, ligand-bound receptors undergo sequential phosphorylation, with progression restarted by ligand unbinding or receptor degradation. High-affinity complexes are kinetically sorted toward degradation-prone states, while low-affinity complexes are sorted towards inactivated states, both limiting signaling. As a result, network activity is maximized for ligands with intermediate affinities. This mechanism explains paradoxical experimental observations in receptor tyrosine kinase (RTK) signaling, including non-monotonic relationships between ligand affinity, kinase activity, and signaling output. Given the ubiquity of multi-site phosphorylation and ligand-induced degradation across signaling pathways, we propose that {\it kinetic sorting} provides a general non-equilibrium strategy for ligand discrimination in cellular networks.