The magnetization and spin transport of asymmetric zigzag-edge graphene nanoribbons, terminated by hydrogen on one edge while unterminated on the other edge, were investigated by a combination of first-principles calculations and a tight-binding approach. At the unterminated edge, a spin-polarized sigma edge state of minority spin appears near the Fermi level and contributes to spin transport. This state enters the band gap for ribbon widths of less than 15 chains, dominating the spin-polarized current. This indicates the importance of the sigma edge states in the design of spintronic devices using graphene nanoribbons. We also examined the case where the 'unterminated' edge is partially terminated by hydrogen.