Thermal MeV neutrino emission from core-collapse supernovae offers a unique opportunity to probe physics beyond the Standard Model in the neutrino sector. The next generation of neutrino experiments such as Hyper-Kamiokande can detect over 10000 neutrinos in the event of a Galactic supernova. As supernova neutrinos propagate to Earth, they may interact with the local dark matter via a hidden vector mediator. Neutrinos interacting with dark matter are delayed with respect to the initial neutrino signal. We show that for 1 keV - 100 keV dark matter, the presence of dark matter-neutrino interactions leads to delays of O(10000000 s). The absence or presence of this feature in the light curve of MeV neutrinos from supernovae allows us to probe a parameter space that is not accessible by dark matter direct detection methods.