The transmission and reflection properties of lossy structures involving left-handed materials with graded permittivity and permeability have been investigated. We present an exact analytical solution to Helmholtz' equation for a lossy case with the graded both real and imaginary parts of permittivity and permeability profile changing according to a hyperbolic tangent function along the direction of propagation. This allows for different loss factors in the two media. The expressions and graphical results for the field intensity along the graded structure are presented. The analytical solution is validated by a dispersive numerical model of lossy metamaterials that uses a transmission line matrix method based on Z-transforms, where a close agreement between the analytic and numerical results is obtained.