Using high-precision photometry, NASA's TESS space mission has discovered many intriguing transiting planet candidates. These discoveries require ground-based follow-up observations, including high-precision Doppler spectroscopy, to rule out false positive scenarios and measure the mass of the transiting planets. In this study, we present an intensive Doppler follow-up campaign of the TESS object of interest TOI-2427, carried out with the High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph to determine the mass of the previously validated transiting planet (TOI-2427 b) and search for additional orbiting companions. By analyzing TESS transit photometry alongside our HARPS radial velocity measurements, we spectroscopically confirmed the transiting planet TOI-2427 b, which orbits its host star every ∼ 1.3 d. We also discovered the presence of a second non-transiting planetary companion with an orbital period of ∼ 5.15 d, which is very close to four times the orbital period of the inner transiting planet. We found that TOI-2427 b is a short-period, high-density super-Earth with a mass of Mb = 5.69‑0.50+0.51 M⊕ and a radius of Rb = 1.64‑0.11+0.12 R⊕, implying a mean density of ρb = 7.1‑0.4+0.8 g cm‑3. Its interior seems to be composed of a predominantly iron core and a silicate mantle and crust. Despite its high density, it is unlikely that TOI-2427 b can sustain any atmosphere composed of lighter gases; however, it could still retain heavier gases. The outer non-transiting planet TOI-2427 c has a minimum mass of Mc sin ic = 6.46‑0.78+0.79 M⊕. Assuming that TOI-2427 b and c are coplanar, a statistical analysis suggests that planets with a mass of ∼ 6.5 M⊕ tend to have radii around 2.7‑0.8+1.1 R⊕. This would place TOI-2457 c near the sub-Neptune regime, while also leaving open the possibility of it being a super-Earth.