We investigate the entropy generation characteristics of a non-Newtonian fluid in a narrow fluidic channel under electrokinetic forcing, taking the effect of conjugate heat transfer into the analysis. We use power-law model to describe the non-Newtonian fluid rheology, in an effort to capture the essential thermohydrodynamics. We solve the conjugate heat transfer problem in an analytical formalism using the thermal boundary conditions of third kind at the outer surface of the walls. We bring out the alteration in the entropy generation behavior as attributable to the rheology-driven alteration in heat transfer, coupled with nonlinear interactions between viscous dissipation and Joule heating originating from electroosmotic effects. We unveil optimum values of different parameters, including both the geometric as well as thermophysical parameters, which lead to the minimization of the entropy generation rate in the system. We believe that the inferences obtained from the present study may bear far ranging consequences in the design of various cooling and heat removal devices/systems, for potential use in microscale thermal management.