In this paper, we present a thorough study of the T Tauri system MO Lup. We report on its previously unrecognized triple spectroscopic nature. Our study is mainly based on high-resolution spectroscopic monitoring. The elements of the spectroscopic orbits are derived from a detailed analysis of radial velocity data for the short-period spectroscopic binary (SB2) components, revealing the existence of a long-term variation in the systemic radial velocity due to a third body orbiting the SB2 system with a period longer than the 5.25-yr span of our data. The estimated stellar parameters, combined with the high lithium content, allow us to confirm the pre-main-sequence (PMS) status of the system. In addition, variability is observed in both the line intensities and profiles of the main emission features, in particular the Hα line, indicative of an intense chromospheric activity in the low-mass SB2 components, possibly suggesting a modulation with the orbital phase. The preliminary orbital solution for the hierarchical triple system makes this object a very suitable target for follow-up observations with the Very Large Telescope Interferometer, for the determination of precise masses of the components that will allow constraints on PMS evolutionary models. Based on observations carried out at the European Southern Observatory at La Silla, Chile in programmes 65.I-0089, 67.C-0155, 68.C-0292, 69.C-0207, 70.C-0163, 72.C-0393, 73.C-0355 and 75.C-399. E-mail: mesposito [at] sa.infn.it (mesposito[at]sa[dot]infn[dot]it)