Six complexes with the general formula Cu­(acylthioureato)­(PPh3)2 were synthesized and characterized using spectroscopic techniques (IR, UV/visible, and 1D and 2D NMR), mass spectrometry, elemental analysis, and X-ray diffraction. Interpretation of the in vitro cytotoxicity data of Cu­(I) complexes took into account their stability in cell culture medium. DFT calculations showed that NMR properties, such as the shielding of carbon atoms, are affected by relativistic effects, supported by the ZORA Hamiltonian in the theoretical calculations. Additionally, the calculation of the energies of the frontier molecular orbitals predicted that the structural changes of the acylthiourea ligands did not cause marked changes in the reactivity descriptors. All complexes were cytotoxic to the evaluated tumor cell lines MDA-MB-231 (triple-negative breast cancer, TNBC), MCF-7 (breast cancer), and A549 (lung cancer). In the MDA-MB-231 cell line, complex 1 significantly altered the cytoskeleton of the cells, reducing the density and promoting the condensation of F-actin filaments. In addition, the compound caused an increase in the percentage of cells in the fragmented DNA region (sub-G0) and induced cell death via the apoptotic pathway starting at the IC50 concentration. Taken together, the results show that complex 1 has cytotoxic and apoptotic effects on TNBC cells, which is a cell line originating from an aggressive, difficult-to-treat breast cancer.