Two copper(II) complexes (Cu-L1, Cu-L2) derived from 2,3-substituted quinazolinone Schiff base ligands (L1, L2) were prepared to examine their anticancer activity. Compounds were characterized using various spectroscopic methods (FTIR, NMR, UV-vis) and quantum-chemical calculations. The biological effects of Cu(II) complexes bearing quinazolinone scaffolds were evaluated on two cancers’ cell lines (breast—MCF-7 and lung—A549), as well as on untransformed cells (keratinocytes—HaCaT). Copper complexes were highly cytotoxic, with IC50 in the low micromolar range, while the quinazoline ligands L1 and L2 remained inactive in inhibiting cell proliferation. Antioxidant activity was investigated in the model systems using DPPH and FRAP assays. The Cu-L1 and Cu-L2 complexes exhibited enhanced DPPH free radical scavenging efficiency compared to the L1 and L2 ligands, but their reducing ability was comparable to that of the free ligands. Evaluation of oxidative stress in vitro carried out by staining cells with various ROS-specific indicators showed reduced production of superoxide anion radical and hydrogen peroxide after treatment of cells with copper complexes. Such a negative impact on ROS formation in cells can lead to cellular redox imbalance and consequent cell death, among others, by inducing apoptosis and/or necrosis, depending on the copper complex used. We hypothesize that the high cytotoxic activity of the investigated copper complexes is apparently the result of multiple mechanisms of action, and the imbalance in the cellular antioxidant system partly contributes to the overall cytotoxic effect.