We report the discovery of a warm Neptune and a hot sub-Neptune transiting TOI-421 (BD-14 1137, TIC 94986319), a bright (V = 9.9) G9 dwarf star in a visual binary system observed by the TESS space mission in Sectors 5 and 6. We performed ground-based follow-up observations — comprised of LCOGT transit photometry, NIRC2 adaptive optics imaging, and FIES, CORALIE, HARPS, HIRES, and PFS 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 5 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-421 b, has an orbital period of Pb ~ 5 days, a mass of Mb ~ 7 ME and a radius of Rb ~ 2.7 RE, whereas the outer warm Neptune, TOI-421 c, has a period of Pc ~ 16 days, a mass of Mc ~ 16 ME, a radius of Rc ~ 5 RE and a density of ρc = 0.685 g/cm3. With its characteristics the inner planet (ρb = 2.05 g/cm3) is placed in the intriguing class of the super-puffy mini-Neptunes. TOI-421 b and c are found to be well suitable for atmospheric characterization. Our atmospheric simulations predict significant Lyα transit absorption, due to strong hydrogen escape in both planets, and the presence of detectable CH4 in the atmosphere of TOI-421 c if equilibrium chemistry is assumed.