This paper uses eigenexpansion technique to describe electro-osmotic effect on unsteady intrusion of a viscous liquid driven by capillary action in a narrow channel. It shows how the dynamics can be manipulated by imposing an electric field along the flow direction in the presence of free charges. Similar manipulation can generate controlled transiency in motion of a complex fluid in a tube by nondestructive forcing leading to efficient rheological measurement. Existing theories analyze similar phenomena by accounting for all involved forces among which the viscous contribution is calculated assuming a steady velocity profile. However, if the transport is strongly transient, a new formulation without an underlying quasi-steady assumption is needed for accurate prediction of the time-dependent penetration. Such rigorous mathematical treatment is presented in this paper where an eigenfunction expansion is used to represent the unsteady flow. Then, a system of ordinary differential equations is derived from which the unknown time-dependent amplitudes of the expansion are determined along with the temporal variation in encroached length. The outlined methodology is applied to solve problems with both constant and periodically fluctuating electric field. In both cases, simplified and convenient analytical models are constructed to provide physical insight into numerical results obtained from the full solution scheme. The detailed computations and the simpler reduced model corroborate each other verifying accuracy of the former and assuring utility of the latter. Thus, the theoretical findings can render a new rheometric technology for effective determination of fluid properties.