We report the discovery of a warm Neptune and a hot sub-Neptune transiting TOI-421 (BD-141137, TIC94986319), a bright (V=9.9) G9 dwarf star in a visual binary system observed by the Transiting Exoplanet Survey Satellite (TESS) space mission in Sectors 5 and 6. We performed ground-based follow-up observations-comprised of Las Cumbres Observatory Global Telescope transit photometry, NIRC2 adaptive optics imaging, and FIbre-fed Echelle Spectrograph, CORALIE, High Accuracy Radial velocity Planet Searcher, High Resolution Echelle Spectrometer, and Planet Finder Spectrograph high-precision Doppler measurements-and confirmed the planetary nature of the 16 day transiting candidate announced by the TESS team. We discovered an additional radial velocity signal with a period of five days induced by the presence of a second planet in the system, which we also found to transit its host star. We found that the inner mini-Neptune, TOI-421b, has an orbital period of Pb=5.19672{+/-}0.00049days, a mass of Mb=7.17{+/-}0.66M{Earth}, and a radius of Rb=2.68_-0.18_^+0.19^R{Earth}, whereas the outer warm Neptune, TOI-421c, has a period of Pc=16.06819{+/-}0.00035days, a mass of Mc=16.42_-1.04_^+1.06^M{Earth}, a radius of Rc=5.09_-0.15_^+0.16^R{Earth}, and a density of {rho}c=0.685_-0.072_^+0.080^g/cm^3^. With its characteristics, the outer planet ({rho}c=0.685_-0.072_^+0.080^g/cm^3^) is placed in the intriguing class of the super-puffy mini-Neptunes. TOI-421b and TOI-421c are found to be well-suited for atmospheric characterization. Our atmospheric simulations predict significant Ly{alpha} transit absorption, due to strong hydrogen escape in both planets, as well as the presence of detectable CH4 in the atmosphere of TOI-421c if equilibrium chemistry is assumed.