We study the influence of decaying perturbations on autonomous oscillatory systems in a plane under the assumption that the perturbations preserve the equilibrium state of the limit system, oscillate with asymptotically constant frequency, and satisfy the nonresonance condition. We discuss the long-term behavior of the perturbed trajectories in a neighborhood of the equilibrium state. We describe conditions on the perturbation parameters that guarantee preservation or loss of stability of the equilibrium. The results are illustrated by an example of decaying perturbations of the Duffing oscillator.