In the present work, we aim to develop a mathematical model for capillary filling dynamics of electromagnetohydrodynamic flow of non-Newtonian fluids. An axially applied electric field and a transverse magnetic field are considered to elucidate the electromagnetohydrodynamic transport through the micro capillary. Assuming a non-Newtonian power-law obeying fluids, we analyse transient evolution of the electromagnetohydrodynamic capillary positions by considering the magnitude of the total force balance via finite volume based numerical formalism. We have highlighted the various rheological regimes in the horizontal capillary through a scaling analysis. For the Newtonian fluids, corresponding inviscid linear Washburn regime is also analysed and compared with the power-law obeying fluids. Furthermore, we have also derived a closed form analytical expressions for the electromagnetohydrodynamic velocity, pressure gradient and transient evolution of the capillary positions by using couple stress parameter model to characterize the fluid rheological behaviours. We perform a comparison test of the coupled stress parameter model with the results from the literature for the similar set of fluid rheological parameters. The comparison results are found to be in good agreement.