Copper(II) complexes with 1,1,1-trifluoro-4-(4-methoxyphenyl)butan-2,4-dione (HL1) were synthesized and characterized by elemental analysis, FT-IR spectroscopy, and single crystal X-ray diffraction. The biological properties of HL1 and cis-[Cu(L1)2(DMSO)] (3) were examined against Gram-positive and Gram-negative bacteria and opportunistic unicellular fungi. The cytotoxicity was estimated towards the HeLa and Vero cell lines. Complex 3 demonstrated antibacterial activity towards S. aureus comparable to that of streptomycin, lower antifungal activity than the ligand HL1 and moderate cytotoxicity. The bioactivity was compared with the activity of compounds of similar structures. The effect of changing the position of the methoxy group at the aromatic ring in the ligand moiety of the complexes on their antimicrobial and cytotoxic activity was explored. We propose that complex 3 has lower bioavailability and reduced bioactivity than expected due to strong intermolecular contacts. In addition, molecular docking studies provided theoretical information on the interactions of tested compounds with ribonucleotide reductase subunit R2, as well as the chaperones Hsp70 and Hsp90, which are important biomolecular targets for antitumor and antimicrobial drug search and design. The obtained results revealed that the complexes displayed enhanced affinity over organic ligands. Taken together, the copper(II) complexes with the trifluoromethyl methoxyphenyl-substituted β-diketones could be considered as promising anticancer agents with antibacterial properties.