The non-Newtonian fluid can be considered as a third-grade fluid with variable viscosity. In this case, the rate of fluid strain can be formulated using the third-grade fluid analogy. In the present study, entropy generation due to non-Newtonian fluid flow in a pipe is investigated. A third-grade fluid with variable viscosity is accommodated in the analysis. Analytical solutions for velocity and temperature distributions are presented, and an entropy generation number is computed for different non-Newtonian parameters, viscosity parameters, and Brinkman numbers. It is found that increasing the non-Newtonian parameter lowers the entropy generation number. This is more pronounced in the region close to the pipe wall. Increasing the viscosity parameter and Brinkman number enhances the entropy generation number, particularly in the vicinity of the pipe wall.