Seeking self-activating chemical nucleases with potential applications as therapeutic agents, new ternary terpyridine–bipyridine–Cu(II) complexes carrying pendant cyclic amines were developed. After detailed characterization, the nuclease activity of the synthesized compounds was evaluated by using circular plasmid DNA as substrate and analyzing the products by agarose-gel electrophoresis. The new complexes present an impressive plasmid DNA cleaving ability, which triggers double-strand DNA breaks in the absence of any exogenous agents, via an oxidative mechanism. The binding affinity towards duplex DNA was determined using UV-Vis and fluorescence spectroscopic titrations. These studies showed that the tested complexes bind moderately (in the order of 104 M−1) to duplex DNA. The copper complexes displayed high cytotoxicity against ovarian carcinoma A2780 cells (4-fold cisplatin activity), surpassing the resistance on the cisplatin-resistant cell line (A2780cisR) with lower resistance factors. Cellular uptake studies showed that the ternary complexes were able to enter the cell with a significant localization in the cytoskeleton.