The spin-polarized electronic structure and half-metallicity of zigzag graphene nanoribbons (ZGNRs) with asymmetric edge terminations are investigated by using first-principles calculations. It is found that compared with symmetric hydrogen-terminated counterparts, such ZGNRs maintain a spin-polarized ground state with the anti-ferromagnetic configuration at opposite edges, but their energy bands are no longer spin degenerate. In particular, the energy gap of one spin orientation decreases remarkably. Consequently, the ground state of such ZGNRs is very close to half-metallic state, and thus a smaller critical electric field is required for the systems to achieve the half-metallic state. Moreover, two kinds of studied ZGNRs present massless Dirac-fermion band structure when they behave like half-metals.