Due to the magnetoelastic coupling, the magnetic properties of many flexible magnetic films (such as Fe, Co, and Ni) are sensitive to mechanical stress, which deteriorates the performance of flexible magnetoelectronic devices. We show that by stacking Co and Pt alternatively to form multilayers with strong perpendicular magnetic anisotropy (PMA), both magnetic hysteresis and magnetic domain measurements reveal robust PMA against external stress. As the PMA weakens at increased Co thickness, the magnetic anisotropy is vulnerable to external stress. These results were understood based on a micromagnetic model, which suggests that the strength of magnetoelastic anisotropy with respect to initial effective magnetic anisotropy affects the stress-stability of the film. Although the stress coefficient of magnetoelastic anisotropy is enhanced at reduced Co thickness, the concomitant increase of initial effective magnetic anisotropy guarantees a robust PMA against external stress. Our results provide a route to constructing flexible magnetoelectronic devices with enhanced stress stability.