We study the influence of oscillatory perturbations on nonlinear nonisochronous oscillatory systems in the plane. We assume that the perturbation amplitude decays and the frequency is unboundedly increasing in time. We study capture into resonance in the case where the amplitude of the system unboundedly increases and the frequency adjusts to the perturbation frequency. We discuss the existence, stability, and asymptotic behavior of resonance solutions at long times. We propose the technique based on averaging method and construction of the Lyapunov functions. The results obtained are applied to the Duffing oscillator with decaying parametric perturbations.