We investigate the spin- and energy-dependent tunneling through a single organic molecule (CoPc) adsorbed on a ferromagnetic Fe thin film, spatially resolved by low-temperature spin-polarized scanning tunneling microscopy. Interestingly, the metal ion as well as the organic ligand show a significant spin dependence of tunneling current flow. State-of-the-art ab initio calculations including also van der Waals interactions reveal a strong hybridization of molecular orbitals and substrate 3d states. The molecule is anionic due to a transfer of one electron, resulting in a nonmagnetic (S=0) state. Nevertheless, tunneling through the molecule exhibits a pronounced spin dependence due to spin-split molecule-surface hybrid states.