Neutrinos of all flavours with mean energies of $\mathcal{O}(10)$MeV are copiusly produced during a Supernova explosion. Its early stage, the so called neutronization burst, is characterized by the emission of a large amounth of electron neutrinos during the first $\sim 25$ms of the explosion, as a result of a fast neutronization of the stellar nucleus via electron capture on free protons. The presence of this sharp time structure in the electron neutrino flavor time distribution makes this channel a very powerful one, allowing large liquid argon underground detectors to provide precision measurements of the time dependence of the electron neutrino fluxes. By exploiting the time-of-flight delay experienced by massive neutrinos, we derive here a new model-independent constraints on the absolute neutrino mass attainable at the future DUNE far detector from a future supernova collapse in our galactic neighborhood. Under favorable scenarios, we found sub-eV results that are competitive with those expected for laboratory direct neutrino mass searches.