An autonomous system of ordinary differential equations describing nonlinear oscillations on the plane is considered. The influence of non-autonomous perturbations decaying at infinity in time is investigated. Such systems are usually called asymptotically autonomous and arise, in particular, as a result of the reduction of multidimensional autonomous and non-autonomous systems. In this paper, we consider a special class of oscillatory perturbations that satisfy the non-resonance condition and do not vanish at the equilibrium of the unperturbed system. We construct a near-identity transformation that averages the system in the first asymptotic terms at infinity in time, and study the structure of the simplified equations. Under some natural assumptions, we describe possible long-term asymptotic regimes for solutions and analyse their stability by constructing Lyapunov functions. In particular, we show how oscillatory terms of perturbations can break the stability of the system and discuss conditions under which the perturbed system behaves like the corresponding unperturbed autonomous system or has new stable regimes. The results obtained are applied to some examples of oscillatory systems with time-decaying oscillatory perturbations.