The O I 777 nm triplet is a key diagnostic of oxygen abundances in the atmospheres of FGK-type stars; however, it is sensitive to departures from local thermodynamic equilibrium (LTE). The accuracy of non-LTE line formation calculations has hitherto been limited by errors in the inelastic O+H collisional rate coefficients; several recent studies have used the Drawin recipe, albeit with a correction factor S_H that is calibrated to the solar centre-to-limb variation of the triplet. We present a new model oxygen atom that incorporates inelastic O+H collisional rate coefficients using an asymptotic two-electron model based on linear combinations of atomic orbitals, combined with a free electron model based on the impulse approximation. Using a 3D hydrodynamic STAGGER model solar atmosphere and 3D non-LTE line formation calculations, we demonstrate that this physically motivated approach is able to reproduce the solar centre-to-limb variation of the triplet to 0.02 dex, without any calibration of the inelastic collisional rate coefficients or other free parameters. We infer log ϵ_O = 8.69 ± 0.03 from the triplet alone, strengthening the case for a low solar oxygen abundance.