Aims. The Scutum complex in 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. Methods. 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 five red supergiants in the young Scutum cluster RSGC3. Results. Taking advantage of the full YJHK spectral coverage of GIANO in a single exposure, we were able to measure several tens of atomic and molecular lines that were suitable for determining chemical abundances. By means of spectral synthesis and line equivalent width measurements, we obtained abundances of Fe and iron-peak elements such as Ni, Cr, and Cu, alpha (O, Mg, Si, Ca, Ti), other light elements (C, N, F, Na, Al, and Sc), and some s-process elements (Y, Sr). We found average half-solar iron abundances and solar-scaled X/Fe abundance patterns for most of the elements, consistent with a thin-disk chemistry. We found depletion of C/Fe and enhancement of N/Fe, consistent with standard CN burning, and low 12C /13C abundance ratios (between 9 and 11), which require extra-mixing processes in the stellar interiors during the post-main sequence evolution. We also found local standard of rest VLSR = 106 km s-1 and heliocentric Vhel = 90 km s-1 radial velocities with a dispersion of 2.3 km s-1. Conclusions. The inferred radial velocities, abundances, and abundance patterns of RSGC3 are very similar to those previously measured in the other two young clusters of the Scutum complex, RSGC1 and RSGC2, suggesting a common kinematics and chemistry within the Scutum complex.