The aim of this work is to show that is possible to apply the Nonlinear Energy Sink (NES) concept to protect seismically excited eccentric steel structures through Targeted Energy Transfer (TET). We consider, as the primary (linear) system, a small-scale four-storey unsymmetrical-plan building, modeled as a twelve-degree-of-freedom-system, with floors sufficiently rigid so that the frame can reasonably be considered as shear-type and with additional eccentric mass for each floor. To the primary (linear) system, we connect two NESs, which are non-smooth and precisely the vibro-impact devices (VI-NESs), both placed on the top floor. In order to analyze the dynamics of the controlled model (structure with VI-NESs), we study the performance and the robustness of the augmented structure excited by a set of Eurocode8 (EC8) spectrum compliant earthquakes. Our purpose is to check the effectiveness of the VI-NESs to different earthquake excitations, that is, testing that an optimal VI-NES setting computed for a specific earthquake will still produce satisfactory results for the other earthquakes. We show that the nonlinear attachments are capable of engaging in transient resonance with linear modes at arbitrary frequencies by generating a one-way irreversible (on the average) transfer of the energy of vibration from the primary structure to local attachment. There the energy is confined and locally dissipated without "spreading" back to the main structure because of the instantaneous internal resonance. As energy decreases due to damping the conditions for Transient Resonance Capture (TRC) fail and escape from resonance capture takes place.