The furocoumarin backbone is a promising platform for chemical modifications aimed at creating new pharmaceutical agents. However, the high level of biological activity of furocoumarins is associated with a number of negative effects. For example, some of the naturally occurring ones and their derivatives can show genotoxic and mutagenic properties as a result of their forming crosslinks with DNA molecules. Therefore, a particularly important area for the chemical modification of natural furocoumarins is to reduce the negative aspects of their bioactivity. By studying a group of 21 compounds—1,2,3-triazolyl modified derivatives of furocoumarin and peucedanin—using the SOS chromotest, the Ames test, and DNA-comet assays, we revealed modifications that can neutralize the structure's genotoxic properties. Theoretical aspects of the interaction of the compound library were studied using molecular modeling and this identified the leading role of the polyaromatic molecular core that takes part in stacking-interactions with the pi-systems of the nitrogenous bases of DNA.