In this paper, a plasmonic perfect absorber based on metal-insulator-graphene-metal structure is proposed as a tunable sensor. The absorption spectrum of this absorber is calculated using the 3D finite difference time domain (FDTD) numerical method for perpendicular incidence of a plane wave light within the 25–65-µm wavelength range. According to the simulation results, a resonance peak is observed in the output absorption spectrum of the device, which is significantly blue-shifted by applying a gate bias voltage to the graphene nano-strips in the structure and increasing the chemical potential of graphene from 150 to 300 meV with 50-meV steps. In another step, in order to investigate the sensing capability, analytes with different refractive indices are poured on the upper surface of the structure, in which the resonance peak is red-shifted with increasing refractive index of the analyte. The sensitivity of this resonance peak is obtained from the relation of S = Δλ/Δn equal to 10,400 nm/RIU, which indicates the excellent ability of the device to detect analytes with a low refractive index difference.