Silver compounds are mainly studied as antimicrobial agents, but they also have anticancer properties, with the latter, in some cases, being better than their gold counterparts. Herein, we analyse the first example of a new Ag(I)-biscarbene that can bind non-canonical structures of DNA, more precisely G-quadruplexes (G4), with different binding signatures depending on the type of G4. Moreover, we show that this Ag-based carbene binds the i-motif DNA structure. Alternatively, its Au(I) counterpart, which was investigated for comparison, stabilises mitochondrial G4. Theoretical in silico studies elucidated the details of different binding modes depending on the geometry of G4. The two complexes showed increased cytotoxic activity compared to cisplatin, overcoming its resistance in ovarian cancer. The binding of these new drug candidates with other relevant biosubstrates was studied to afford a more complete picture of their possible targets. In particular, the Ag(I) complex preferentially binds DNA structures over RNA structures, with higher binding constants for the non-canonical nucleic acids with respect to natural calf thymus DNA. Regarding possible protein targets, its interaction with the albumin model protein BSA was also tested.