In this study, the difference between laminar fast transient flow of shear-thinning liquids and that of Newtonian liquids under similar conditions is numerically studied. Since the literature appears to lack fast transient flow investigation of non-Newtonian fluids, this work addresses features of those flows. In this way, the Newton–Kantorovich method is implemented to linearize nonlinear shear stress term available in the characteristic equations. The verification and validation of the solution are carried out in detail. The results show that the non-Newtonian behavior of fluids has significant influence on the velocity and shear stress profiles and also on the magnitude of pressure head and wall shear stress.