Injecting spins into nonmagnetic molecular devices has attracted much attention in molecular spintronics. Herein, we propose a novel strategy to introduce magnetism into a single benzene molecule coupled with two armchair graphene nanoribbons (AGNR) electrodes, where the ends of two AGNR electrodes are cut into zigzag-edge triangular graphenes (ZTGs). The spin-dependent transport properties of the molecular junction are investigated by using the density functional theory (DFT) combined with the non-equilibrium Green’s function (NEGF) method. The analyses of the spin-dependent projected density of states and the net spin density distribution of the scattering region reveal that the intrinsic magnetism of the ZTGs is weakened, owing to spin transfer from ZTGs to AGNR electrodes and the benzene molecule. More interestingly, the attenuated intrinsic magnetism of the ZTGs can still contribute to a significant spin transport of the molecular junction. Transport calculations show that in the parallel spin configuration, a large spin polarization of nearly 90% current is obtained. However, the spin polarization of current is reversed in antiparallel spin configuration. Positive or negative tunneling magnetoresistance (TMR) can be modulated by bias voltage. A TMR up to 53% is obtained in the device. The results are further analyzed from the transmission spectra and local density of states. This work presents a promising potential applications of the ZTGs in the field of molecular spintronics, which can contribute to the design of graphene nanoribbons based molecular spintronic devices.