For various industrial applications, it is necessary to investigate the behavior of non-Newtonian fluids, specifically the power-law fluids, within corrugated channels. The exact expressions are derived relating the volume flow rate and change in pressure across the channel length for the incompressible, laminar, and viscous flow, focusing on the flow of power-law fluids. The analytical method considers the five different configurations of 2D planar converging–diverging corrugated channels such as linear wedge, parabolic wedge, hyperbolic profile, hyperbolic cosine profile, and the sinusoidal. The pressure-flow rate relations are obtained under the lubrication approximation. It is noticed that when the flow viscosity is taken as a constant, the method approaches the Newtonian flow physics. For the given specific maximum and minimum channel heights and fluid properties such as the power index, and pressure drop, it is observed that the flow rate depends on the shape of the channels. Additionally, as the value of power index increases, i.e., as the fluid transitions from shear-thinning to shear-thickening, the flow rate decreases. Furthermore, it is noticed that the volume flow rate is linearly proportional to the change in pressure across all channel types. The present method can be utilized for several types of fluids ranging from shear thinning to shear-thickening and specific channel profiles. In cases where complex mathematical and practical considerations pose a challenge in obtaining analytical expressions, the present investigation provides a strong frame of reference for obtaining accurate numerical results.