The effect of a floating elastic membrane on a gravity-driven flow down an inclined porous plane is examined. The present work extends the earlier work of Anjalaiah et al. [Phys. Fluid, Vol. 25, 022101 (2013)] by including the thin membrane and excluding the insoluble surfactant at the top surface of the flow. The membrane tension emergence from the elasticity, is used to monitoring the surface tension on the top surface of the fluid layer. The fluid flow below the elastic membrane is governed by the Navier–Stokes equations and flow through the porous medium is governed by the Darcy–Brickmann equations. Using the normal mode analysis, two Orr–Sommerfeld systems are derived for each layers and solved by using the spectral collocation method to capture the linear instability. Further, the temporal growth rate and marginal stability curve are analyzed for different set of structural parameters. In general, the instability of free surface flow is more due to the presence of the unstable Yih mode. Both, the membrane tension and the uniform mass distribution rate along the length of the membrane assure to stabilize the surface waves by lowering the temporal growth rate of the disturbances. Consequently, the floating elastic membrane and the porosity of the inclined substrate pretend to suppress the unstable Yih mode and stabilize the flow anatomy. Hence, the floating elastic membrane can be disposed for the lethargic control of the free-surface flows' instability. © 2020 Wiley-VCH GmbH