The adsorption of CO on the surface of metals such as Pt(111) is of great interest owing to the industrial importance of the catalytic oxidation of pollutant CO. To date, reliable high-level calculations of this process have not been possible, a situation often referred to as the "CO/Pt(111) puzzle". Standard generalized-gradient-approximation density functional theory approaches fail to capture key details of the binding, such as the location of the adsorption site, while cluster approaches using alternative methods show some but insufficient improvement. Using a new computational methodology combining hybrid density functionals containing non-local Hartree-Fock exchange with periodic imaging plane-wave-based techniques, we demonstrate that key aspects of the adsorption of CO on Pt(111), including the identification of the absorption site and CO frequency change, can now be adequately modeled. The binding is dominated by both CO dative covalent bonding and metal-to-molecule pi back-bonding, effects requiring realistic alignment of both the molecular HOMO and LUMO orbitals with respect to the metal Fermi energy.