Lur'e-type systems with periodic nonlinearities arise in many physical and engineering applications, from the simplest model of a pendulum to large-scale networks of power generators or biological oscillators. Periodic nonlinearities often cause the existence of multiple stable and unstable equilibria, which can lead to presence of "hidden attractors" and other complex phenomena. Many tools of classical nonlinear control, developed for systems with globally stable equilibria, become inapplicable for pendulum-like systems. To study their asymptotic properties, special Lyapunov techniques have been developed based on special periodic Lyapunov functionals. In this paper, we extend this method to address the problem of robustness against uncertain external disturbances. We are primarily interested in the situation, where the disturbance decays at infinity or, more generally, has a finite limit, which enables the disturbed system to have equilibria. A natural question then arises whether asymptotic properties of the system (e.g. the solutions' convergence) are robust against the disturbance. We give a sufficient condition ensuring such a robustness.