A series of six new bis(thiosemicarbazone)copper(I) complexes of the type [Cu(L1–6)2Cl] (1−6) were synthesized and characterized. The complexes adopted trigonal planar ′Y′ shaped geometry coordinating through two thione sulphur atoms of two ligand molecules and one chloride ion. All the complexes intercalatively bind with calf thymus DNA (CT−DNA) as evidenced by spectral and molecular docking studies. The absorption and emission spectral techniques confirmed the strong interaction of the complexes with BSA via static quenching mode. The complexes efficiently cleave pBR322 DNA via hydrolytic pathway, and significantly interact with epidermal growth factor receptor. All the complexes were assessed for their anti‐proliferative activity, in which the complexes 2, 3 and 4 containing methyl, methoxy and hydroxyl groups, respectively, showed significant activity. The complexes induce apoptosis in EAC cells as evidenced by acridine orange (AO)/ethidium bromide (EB), Hoechst 33258 and propidium iodide (PI) staining methods, and cell cycle analysis. Cellular uptake studies revealed the ability of the complexes to go into the cytoplasm and accumulation in the cell nuclei. The complexes are involved in the generation of reactive oxygen species (ROS), mitochondrial mediated and caspase‐dependent apoptosis. Further, using a female Swiss albino mice model, we found that the complexes 2 and 3 inhibited Ehrlich ascites carcinoma (EAC) tumour cell growth in vivo.