We investigated propagation of electromagnetic waves through composite structures with negative refractive index, the popular ”left-handed metamaterials”, for the case when there is a gradient of refractive index. We obtained the exact analytical solutions to the Helmholtz equation valid for arbitrary steepness of the graded interface between the positive and the negative index part. We analyzed the special case of matched impedances of the two constituent materials within the metamaterial composite. We derived analytical expressions for the field intensity, transmission and reflection coefficients and compared them with the results obtained by the numerical simulations using the Finite Element Method. The model allows for arbitrary spectral dispersion and lossy media.