This paper presents an efficient approach for exciting a dielectric resonator antenna (DRA) in the terahertz frequencies by means of a graphene plasmonic dipole. Design and analysis are performed in two steps. First, the propagation properties of hybrid plasmonic onedimensional and two-dimensional structures are obtained by using transfer matrix theory and the finite-element method. The coupling amount between the plasmonic graphene mode and the dielectric wave mode is explored based on different parameters. These results, together with DRA and plasmonic antenna theory, are then used to design a DRA antenna that supports the TEy112 mode at 2.4 THz and achieves a gain (IEEE) of up to 7 dBi and a radiation efficiency of up 70%. This gain is 6.5 dB higher than that of the graphene dipole alone and achieved with a moderate area overhead, demonstrating the value of the proposed structure.