A rigorous analysis of coupled nonlinear equations for third-grade viscoelastic power-law non-Newtonian fluid is presented. Initially, the governing partial differential equations for conservation of energy and momentum are transformed to nonlinear coupled ordinary differential equations using exact similarity transformations which are known as Cattaneo–Christov heat flux model for third-grade power-law fluid. The homotopy analysis method (HAM) is utilized to approximate the systematic solutions more precisely with shear-thickening, moderately shear-thinning, and most shear-thinning fluids. The solution depends on various parameters including Prandtl number, power index, and temperature variation coefficient. A systematic analysis of boundary-layer flow demonstrates the impact of these parameters on the velocity and temperature profiles.