Thin lubrication flows accompanying drainage from curved surfaces surround us (e.g., the drainage of the tear film on our eyes). These draining aqueous layers are normally covered with surface-active molecules that render the free surface viscoelastic. The non-Newtonian character of these surfaces fundamentally alters the dynamics of drainage. We show that increased film stability during drainage can occur as a consequence of enhanced surface rheology. Increasing the surfactant layer viscosity decreases the rate of drainage; however, this retarding influence is most pronounced when the insoluble surfactant layer has significant elasticity. We also present a simple theoretical model that offers qualitative support to our experimental findings.