A tunable terahertz metamaterial absorber with transmission characteristics is proposed and numerically studied. The proposed four-layer structure of the absorber is comprised of a patterned gold resonator, a patterned graphene layer, a silicon dioxide (SiO₂) substrate, and a bottom gold layer. Unlike the other common designs reported in literature that block the transmission of electromagnetic waves using a bottom metal plane, the proposed design allows for the transmission of electromagnetic waves by using a slotted gold-metallic ground plane. Through simulation analysis it is found that high absorption of 97.3% occurs at 0.9512 THz while a transmission peak of 90% occurs at 0.440 THz, thus proving the high absorption and non-zero transmission in a small structure. The impedance is 420 Ω at 0.9512 THz which is close to the characteristic impedance of air 377 Ω indicating the effective coupling of the electromagnetic waves to the absorber with minimized reflection at the air interface. The ability to tune the resonance response dynamically with graphene is achieved by tuning the chemical potential and relaxation time. The absorber also shows polarization-insensitive performance and large-angle stability as large as 70° for both TE and TM modes. Apart from this, it is shown that the proposed metamaterial absorber also has the ability of refractive-index sensing with a sensitivity of 0.1488 THz / RIU.