The inner disk of the Galaxy has a number of young star clusters dominated by red supergiants that are heavily obscured by dust extinction and observable only at infrared wavelengths. These clusters are important tracers of the recent star formation and chemical enrichment history in the inner Galaxy. During the technical commissioning and as a first science verification of the GIANO spectrograph at the Telescopio Nazionale Galileo, we secured high-resolution (R~50,000) near-infrared spectra of three red supergiants in the young Scutum cluster RSGC2. Taking advantage of the full YJHK spectral coverage of GIANO in a single exposure, we were able to identify several tens of atomic and molecular lines suitable for chemical abundance determinations. By means of spectral synthesis and line equivalent width measurements, we obtained abundances of Fe and other iron-peak elements such as V, Cr, Ni, of alpha (O, Mg, Si, Ca and Ti) and other light elements (C, N, Na, Al, K, Sc), and of some s-process elements (Y, Sr). We found iron abundances between half and one third solar and solar-scaled [X/Fe] abundance patterns of iron-peak, alpha and most of the light elements, consistent with a thin-disk chemistry. We found a depletion of [C/Fe] and enhancement of [N/Fe], consistent with CN burning, and low 12C/13C abundance ratios (between 9 and 11), requiring extra-mixing processes in the stellar interiors during the post-main sequence evolution. Finally, we found a slight [Sr/Fe] enhancement and a slight [Y/Fe] depletion (by a factor of <=2), with respect to solar.