The Transiting Exoplanet Survey Satellite, TESS, is currently carrying out an all-sky search for small planets transiting bright stars. In the first year of the TESS survey, a steady progress was made in achieving the mission's primary science goal of establishing bulk densities for 50 planets smaller than Neptune. During that year, the TESS's observations were focused on the southern ecliptic hemisphere, resulting in the discovery of three mini-Neptunes orbiting the star TOI-125, a V = 11.0 K0 dwarf. We present intensive HARPS radial velocity observations, yielding precise mass measurements for TOI-125b, TOI-125c, and TOI-125d. TOI-125b has an orbital period of 4.65 d, a radius of 2.726 ± 0.075 RE, a mass of 9.50 ± 0.88 ME, and is near the 2:1 mean motion resonance with TOI-125c at 9.15 d. TOI-125c has a similar radius of 2.759 ± 0.10 RE and a mass of 6.63 ± 0.99 ME, being the puffiest of the three planets. TOI-125d has an orbital period of 19.98 d and a radius of 2.93 ± 0.17 RE and mass 13.6 ± 1.2 ME. For TOI-125b and d, we find unusual high eccentricities of 0.19 ± 0.04 and 0.17^{+0.08}_{-0.06}, respectively. Our analysis also provides upper mass limits for the two low-SNR planet candidates in the system; for TOI-125.04 (RP = 1.36 RE, P = 0.53 d), we find a 2σ upper mass limit of 1.6 ME, whereas TOI-125.05 (R_P=4.2^{+2.4}_{-1.4} RE, P = 13.28 d) is unlikely a viable planet candidate with an upper mass limit of 2.7 ME. We discuss the internal structure of the three confirmed planets, as well as dynamical stability and system architecture for this intriguing exoplanet system.