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We investigate the rheological properties of a non-Newtonian glass-former liquid within lithographically defined microchannels in the range of temperatures above the vitrification region. The non-Newtonian behavior of the fluid, as evidenced by rotational rheology, is well described by a power law dependence of the viscosity on the shear rate. Taking into account such non-Newtonian character in the equations for the microfluidic motion, we relate the penetration dynamics into capillaries with the liquid rheological properties. The temperature dependence of the viscosity, determined over 1 order of magnitude in the temperature range 286-333 K and for shear rates between 0.07 and 1 [s.sup.-1], can be described by a Vogel--Fulcher--Tamman law, consistent with the fragile nature of the investigated compound. Microfluidics is a promising analytical approach for the investigation of the rheology of non-Newtonian fluids within confined microenvironments.