We developed a microfluidic analogue of the classic Wheatstone bridge circuit for automated, real-time sampling of solutions in a flow-through device format. We demonstrate precise control of flow rate and flow direction in the "bridge" microchannel using an on-chip membrane valve, which functions as an integrated "variable resistor". We implement an automated feedback control mechanism in order to dynamically adjust valve opening, thereby manipulating the pressure drop across the bridge and precisely controlling fluid flow in the bridge channel. At a critical valve opening, the flow in the bridge channel can be completely stopped by balancing the flow resistances in the Wheatstone bridge device, which facilitates rapid, on-demand fluid sampling in the bridge channel. In this article, we present the underlying mechanism for device operation and report key design parameters that determine device performance. Overall, the microfluidic Wheatstone bridge represents a new and versatile method for on-chip flow control and sample manipulation.