When ionizing radiation passes biological matter, a large number of secondary electrons with very low energies (<3 eV) is produced. It is known that such electrons cause an efficient fragmentation of isolated nucleobases via dissociative electron attachment. We present an experimental study of the electron attachment to microhydrated nucleobases. Our novel approach allows significant control over the hydration of molecules studied in the molecular beam. We directly show for the first time that the presence of a few water molecules suppresses the dissociative channel and leads exclusively to formation of intact molecular and hydrated anions. The suppression of fragmentation is ascribed to caging-like effects and fast energy transfer to the solvent. This is in contrast with theoretical prediction that microhydration strongly enhances the fragmentation of nucleobases. The current observation impacts mechanisms of reductive DNA strand breaks proposed to date on the basis of gas-phase experiments.