The isotope ^{98}Tc decays to ^{98}Ru with a half-life of 4.2×10^{6} yr and could have been present in the early Solar System. In this Letter, we report on the first calculations of the production of ^{98}Tc by neutrino-induced reactions in core-collapse supernovae (the ν process). Our predicted ^{98}Tc abundance at the time of solar system formation is not much lower than the current measured upper limit raising the possibility for its detection in the not too distant future. We show that, if the initial abundance were to be precisely measured, the ^{98}Tc nuclear cosmochronometer could be used to evaluate a much more precise value of the duration time from the last core-collapse supernova to the formation of the solar system. Moreover, a unique and novel feature of the ^{98}Tc ν-process nucleosynthesis is the large contribution (∼20%) from charged current reactions with electron antineutrinos. This means that ^{98}Tc becomes a unique new ν-process probe of the temperature of the electron antineutrinos.