We theoretically investigate the light-graphene interactions enabled by a single layer of nonlossy nanorods at near-infrared wavelengths. The sustained Fano-like geometric resonance gives rise to enhanced graphene absorption, e.g., 100%, and adjustable absorption linewidth even to be ultra-narrow, e.g., < 1 nm. The conditions for such graphene absorption enhancement and linewidth sharpening are analytically interpreted within the framework of temporal coupled mode theory for the Fano resonance. The geometric resonance enhanced light-graphene interactions are polarization-sensitive and angle-dependent. Our study offers new possibilities towards designing and fabricating novel opto-electronic devices such as graphene-integrated monochromatic photodetectors and ultra-compact modulators.