Novel complexes of type [Cu(N-N)(dmtp)2(OH2)](ClO4)2·dmtp ((1) N-N: 2,2′-bipyridine; (2) L: 1,10-phenantroline and dmtp: 5,7-dimethyl-1,2,4-triazolo[1,5-a]pyrimidine) were designed in order to obtain biologically active compounds. Complexes were characterized as mononuclear species that crystallized in the space group P-1 of the triclinic system with a square pyramidal geometry around the copper (II). In addition to the antiproliferative effect on murine melanoma B16 cells, complex (1) exhibited low toxicity on normal BJ cells and did not affect membrane integrity. Complex (2) proved to be a more potent antimicrobial in comparison with (1), but both compounds were more active in comparison with dmtp—both against planktonic cells and biofilms. A stronger antimicrobial and antibiofilm effect was noticed against the Gram-positive strains, including methicillin-resistant Staphylococcus aureus (MRSA). Both electron paramagnetic resonance (EPR) and Saccharomyces cerevisiae studies indicated that the complexes were scavengers rather than reactive oxygen species promoters. Their DNA intercalating capacity was evidenced by modifications in both absorption and fluorescence spectra. Furthermore, both complexes exhibited nuclease-like activity, which increased in the presence of hydrogen peroxide.