Most of the energy deposited in cells by ionizing radiation is channeled into the production of abundant free secondary electrons with ballistic energies between 1 and 20 electron volts. Here it is shown that reactions of such electrons, even at energies well below ionization thresholds, induce substantial yields of single- and double-strand breaks in DNA, which are caused by rapid decays of transient molecular resonances localized on the DNA's basic components. This finding presents a fundamental challenge to the traditional notion that genotoxic damage by secondary electrons can only occur at energies above the onset of ionization, or upon solvation when they become a slowly reacting chemical species.