We report the synthesis of five new Cu(I) acylthiourea complexes (C1–C5) bearing the general formula [Cu(L-R)Cl (PPh3)2] [L = monodentate acylthiourea ligand, R = C6H5 (L1), C6H4CH3(o) (L2), C6H4OCH2CH3(p) (L3), C10H7 (L4) or C6H4Cl(p) (L5)]. All the complexes were characterized by analytical and spectroscopic tools. The complexes (C1–C5) exhibited a distorted tetrahedral geometry as inferred from the single crystal X-ray diffraction study. The complexes were subjected to interact with the biomolecules (calf thymus [CT] DNA/bovine serum albumin [BSA]); the one bearing naphthyl substituent (C4) exhibited the highest binding efficacy. Further, anticancer activity of the complexes was studied exclusively against breast cancer cell lines, namely, MCF7, T47D, and MDA MB 231. Complex C4 was found to be highly cytotoxic on the three cancer cell lines with the IC50 values of 0.75, 0.75, and 0.68 μM, respectively. Conveniently, the complexes displayed fourfold less toxicity against the normal MCF10a human breast cells. The ability of complex C4 to induce apoptosis was analyzed by acridine orange/ethidium bromide (AO/EB) and Hoechst 33258 staining assays. Furthermore, it was found that complex C4 induced apoptosis via reactive oxygen species (ROS)-mediated mitochondrial signaling pathway. Confocal fluorescence images of the cells subjected to lyso and mitotracker staining assays revealed that complex C4 was primarily localized on the mitochondria, and finally, Western blot analysis also confirmed the apoptosis induced by complex C4 in the MDA MB 231 cancer cells.