We investigate the transport of immiscible binary fluid layers, constituted by one conducting (top layer fluid) and another non-conducting (bottom layer fluid) fluids in a microfluidic channel under the combined influences of an applied pressure gradient and imposed electric field. We solve the transport equation governing the flow dynamics analytically and obtain the closed-form expressions of the velocity fields. We bring out the alteration in the flow dynamics, mainly attributable to the non-linear interaction between interfacial slip and the electrical double layer effect over small scales as modulated by the applied pressure gradient. In particular, we show the augmentation in the net volume transport rate through the channel, emerging from an intricate competition among electrical forcing, applied pressure gradient and the viscous resistance as modulated by the interfacial slip. We believe that the results of this studymay be of immense consequence for the design of various microfluidic devises, which are often used for the manipulation of two immiscible fluids in different biomedical/biochemical processes.