A graphene-based controllable broadband terahertz metamaterial absorber with transmission band is presented in this paper. It consists of a graphene-SiO₂-frequency selective surface (FSS) sandwich structure. The sinusoidal graphene layer supports continuous plasmonic resonances, forming a broad electric-tuning absorbing band. Bandpass FSS constructs a transmission window outside the absorbing band. The simulation results indicate that the absorption from 0.5 THz to 1 THz can be tuned continuously from 0.4 to 0.9 with angle and polarization independence. A transparent window peaking at 1.65 THz maintains high transmittance over 0.7. The metamaterial absorber has potential applications for detection, stealth, filtering, and electromagnetic compatibility.