Complete, exceptionally-preserved skulls of the Permian lungfish Persephonichthys chthonica gen. et sp. nov. are described. Persephonichthys chthonica is unique among post-Devonian lungfishes in preserving portions of the neurocranium, permitting description of the braincase of a stem-ceratodontiform for the first time. The completeness of P. chthonica permits robust phylogenetic analysis of the relationships of the extant lungfish lineage within the Devonian lungfish diversification for the first time. New analyses of the relationships of this new species within two published matrices using both maximum parsimony and Bayesian inference robustly place P. chthonica and modern lungfishes within dipterid-grade dipnoans rather than within a clade containing Late Devonian 'phaneropleurids' and common Late Paleozoic lungfishes such as Sagenodus. Monophyly of post-Devonian lungfishes is not supported and the Carboniferous-Permian taxon Sagenodus is found to be incidental to the origins of modern lungfishes, suggesting widespread convergence in Late Paleozoic lungfishes. Morphology of the skull, hyoid arch, and pectoral girdle suggests a deviation in feeding mechanics from that of Devonian lungfishes towards the more dynamic gape cycle and more effective buccal pumping seen in modern lungfishes. Similar anatomy observed previously in 'Rhinodipterus' kimberyensis likely represents an intermediate state between the strict durophagy observed in most Devonian lungfishes and the more dynamic buccal pump seen in Persephonichthys and modern lungfishes, rather than adaptation to air-breathing exclusively.