We report on the magnetic properties of small neutral suboxide ConOm (n = 5-18 and m = 0-10, m ≤ n) clusters produced by laser vaporisation and gas aggregation. Their magnetism is probed experimentally by means of Stern-Gerlach magnetic deflection. The results imply that the cobalt atoms couple ferromagnetically not only in pure Con clusters, as known from previous investigations, but also in their oxidized counterparts. It was found that the magnetic moment per cobalt atom is mostly enhanced in the oxide clusters with respect to the pure cobalt clusters and generally increases with the oxygen content in the studied composition range. The spin magnetism of selected clusters is also investigated by density functional theory (DFT) calculations. The computations allow to attribute the effect of oxidation on the magnetic response of the ConOm clusters to electron transfer from the cobalt 3d and 4s valence orbitals to oxygen. The cobalt 3d levels preferentially donate electrons of minority spin, but both spin orientations are involved in the transfer of cobalt 4s electrons.