Novel metamaterial refractive index sensor based on impedance matching concept is proposed to achieve an ultra-narrowband absorption response in terahertz (THz) frequencies. To this end, the recently proposed circuit model of the periodic arrays of graphene disks has been exploited to model the graphene layer as circuit components while a developed transmission line method has been used to provide a circuit model for the whole structure. As a result, a perfect THz absorption response with absorptivity over 99.5% is achieved in desired resonance frequency. Two sensor structures at 3 and 6 THz have been designed as numerical examples, in which the frequency sensitivity of the structures is achieved as 0.834 and 1.57 THz per refractive index while the figure-of-merit of 11.75 and 24.5 are obtained, respectively. Its particular structure is polarization insensitive, further wide-angle absorptivity for both transverse electric and transverse magnetic waves is obtained. To verify the accuracy and validity of the presented circuit model approach, full-wave numerical modeling is performed.