In this Letter we report on effects of nucleon-nucleon correlations probed in nucleon transfer reactions with heavy ions. We measured with high efficiency and resolution a complete set of observables for neutron transfer channels in the ^{206}Pb+^{118}Sn system employing a large solid angle magnetic spectrometer, which allowed us to study a wide range of internuclear distances via a detailed excitation function. The coupled channel theory, based on an independent particle transfer mechanism, follows the experimental transfer probabilities for one- and two-neutron pick-up and stripping channels. The experimental two-neutron transfer cross sections indicate that in reactions between pair-vibrational (closed shell) and pair-rotational (open shell) nuclei, correlations manifest via pair-addition and pair-removal modes, which constitute one of the elementary modes of excitations in nuclei.