Biological valves control the directionality of oscillatory or fluctuating internal flows by deforming (reversibly) to stop flow. To uncover the physical basis for flow rectification in these systems, we introduce a simple physical mimic of the mitral heart-valve that is shaped like a soft conical shell in an impinging flow. Combining ex vivo porcine-heart and in vitro conical valve experiments, we show how noise-activated buckling deformations serve to rectify the flow. We quantitatively explain our observations using a minimal elastohydrodynamic theory for valve closure that leads to a dynamical bifurcation driven by stochastic hydrodynamic forces. Our theory also suggests that valve collapse can be triggered on demand, which we corroborate using experiments. Together, these results suggest a design principle for the efficient operation of soft valves.