We investigate TM-wave propagation in a hollow waveguide with a graded dielectric layer, described using a hyperbolic tangent function. General formulae for the electric field components of the TM-waves, applicable to hollow waveguides with arbitrary cross sectional shapes, are presented. We derive the analytical results for the reflection and transmission coefficients valid for waveguides of arbitrary cross sectional shapes. Thereby, we show that the obtained reflection and transmission coefficients are in exact asymptotic agreement with those obtained for a very thin homogeneous dielectric layer using mode-matching and cascading. The proposed method is tractable since it gives analytical results that are directly applicable without the need of mode-matching, and it has the ability to model realistic, smooth transitions.