To help in the efficient design of fluid flow in electroosmotic pumps, electroosmotic flow of non-Newtonian fluid through porous polymer membrane at high zeta potentials is studied by mainly evaluating the total flow rate at different physical parameters. Non-Newtonian fluid is represented by the power-law model and the porous polymer membrane is considered as arrays of straight cylindrical pores. The electroosmotic flow of non-Newtonian fluid through a single pore is studied by solving the complete Poisson-Boltzmann equation and the modified Cauchy momentum equation. Then assuming the pore size distribution on porous polymer membrane obeys Gaussian distribution, the performance of electroosmotic pump operating non-Newtonian fluid is evaluated by computing the total flow rate of electroosmotic flow through porous polymer membrane as a function of flow behavior index, geometric parameters of porous membrane, electrolyte concentration, applied voltage, and zeta potential. It is found that enhancing zeta potential and bulk concentration rather than the applied voltage can also significantly improve the total flow rate in porous polymer membrane, especially in the case of shear thinning fluid.