Quantum-gravity effects might generate Lorentz invariance violation by the interaction of energetic particles with the foamy structure of the space-time. As a consequence, particles may not travel at the universal speed of light. We propose to constrain the Lorentz invariance violation for energetic neutrinos exploiting the νe neutronization burst from the next galactic supernova (SN). This prompt signal is expected to produce a sharp peak in the SN νe light curve with a duration of ∼25 ms. However, the presence of the energy-dependent Lorentz invariance violation would significantly spread out the time structure of this signal. We find that the detection the SN νe burst from a typical galactic explosion at d=10 kpc in a Mton-class water Cerenkov detector would be sensitive to a quantum-gravity mass scale MQG∼1012 GeV (2×105 GeV) for the linear (quadratic) energy dependence of the Lorentz invariance violation. These limits are valid for both super and subluminal neutrino velocity and are also independent of the neutrino mass hierarchy.