This paper presents analytical and numerical studies of electromagnetic wave propagation through an interface between a regular right-handed material (RHM) and a left-handed metamaterial (LHM). The interface is graded along the direction perpendicular to the boundary plane between the two materials, chosen to be the x-direction. The permittivity ε(ω, x) and permeability μ(ω, x) are chosen to vary according to hyperbolic tangent functions. We show that the field intensities for both TE- and TM-cases satisfy the same differential equations, and we obtain remarkably simple exact analytical solutions to Helmholtz' equations for lossy media. The obtained exact analytical results for the field intensities along the graded RHM-LHM composite are in line with the expected properties of RHM-LHM structures. Finally, we perform a numerical study of the wave propagation over an impedance-matched graded RHM-LHM interface, using the software COMSOL Multiphysics, and obtain an excellent agreement between the numerical simulations and analytical results. The results obtained in the present paper are not limited to any particular application, and are generally useful for all cases of wave propagation over impedance-matched two- and three-dimensional interfaces between RHM and LHM media. The advantage of the present method is that it can model smooth realistic material transitions, while at the same time including the abrupt transition as a limiting case. Furthermore, unlike previously existing solutions, the interface width is included as a parameter in the analytical solutions in a very simple way. This enables the use of the interface width as an additional degree of freedom in the design of practical RHM-LHM interfaces.