We report spin-polarized density functional calculations of ferromagnetic properties for a series of ZnO clusters and ZnO solid containing one or two substitutional carbon impurities. We analyze the eigenvalue spectra, spin densities, molecular orbitals, and induced magnetic moments for ZnC, Zn2C, Zn2OC, carbon-substituted ZnnOn (n = 3-10, 12) clusters and the bulk ZnO. The results show that the doping induces magnetic moment of similar to 2 mu(B) in all the cases. All systems with two carbon impurities show ferromagnetic interaction, except when carbon atoms share the same zinc atom as the nearest neighbor. This ferromagnetic interaction is predominantly mediated via pi-bonds in the ring structures and through pi- and sigma-bonds in the three-dimensional structure. The calculations also show that the interaction is significantly enhanced in the solid, bringing out the role of dimensionality of the Zn-O network connecting two carbon atoms.