A metal–graphene metamaterial device exhibiting a tunable, electromagnetically induced transparency (EIT) spectral response at terahertz frequencies is investigated. The metamaterial structure is composed of a strip and a ring resonator, which serve as the bright and dark mode to induce the EIT effect. By employing the variable conductivity of graphene to dampen the dark resonator, the response frequency of the device shifts dynamically over 100 GHz, which satisfies the convenient post-fabrication tunability requirement. The slow-light behavior of the proposed device is also analyzed with the maximum group delay of 1.2 ps. The sensing performance is lastly studied and the sensitivity can reach up to 100 GHz/(RIU), with a figure of merit (FOM) value exceeding 4 ${\mathrm{RIU}}^{-1}$. Therefore, the graphene-based metamaterial provides a new miniaturized platform to facilitate the development of terahertz modulators, sensors, and slow-light applications.