Controlled pumping of fluids through microfluidic networks is a critical unit operation ubiquitous to lab-on-a-chip applications. Although there have been a number of studies involving the creation of passive flows within lab-on-a-chip devices, none has shown the ability to create temporally stable flows for periods longer than several minutes. Here a passive pumping approach is presented in which a large pressure differential arising from a small, curved meniscus situated along the bottom corners of an outlet reservoir serves to drive fluid through a microfluidic network. The system quickly reaches steady-state and is able to provide precise volumetric flow rates for periods lasting over an hour. A two-step mathematical model provides accurate predictions of fluid and mass transport dynamics in these devices, as validated by particle tracking in laboratory systems. Precise flow rates spanning an order of magnitude are accomplished via control of the microchannel and outlet reservoir dimensions. This flow mechanism has the potential to be applied to many micro-total analytical system devices that utilize pressure-driven flow; as an illustrative example, the pumping technique is applied for the passive generation of temporally stable chemical gradients.