We utilize elastic deformations via mechanical actuation to control and direct fluid flow within a flexible microfluidic device. The device consists of a microchannel with a flexible arch prepared by the buckling of a thin elastic film. The deflection of the arch can be predicted and controlled using the classical theory of Euler buckling. The fluid flow rate is then controlled by coupling the elastic deformation of the arch to the gap within the microchannel, and the results compared well with analytical predictions from a perturbation calculation and numerical simulations. We demonstrate that placement of these flexible valves in series enables directed flow towards regions of externally applied mechanical stress. The simplicity of the experimental approach provides a general design for advanced functionality in portable microfluidics, self-healing devices, and in situ diagnostics.