Aseries of novel 1,4-disubstituted 1,2,3-triazoles were synthesized from an (R)-carvone terminal alkyne derivative via a Cu (I)-catalyzed azide–alkyne cycloaddition reaction using CuSO4,5H2O as the copper (II) source and sodium ascorbate as a reducing agent which reduces Cu (II) into Cu (I). All the newly synthesized 1,2,3-triazoles 9a–h were fully identified on the basis of their HRMS and NMR spectral data and then evaluated for their cell growth inhibition potential by MTS assay against HT-1080 fibrosarcoma, A-549 lung carcinoma, and two breast adenocarcinoma (MCF-7 and MDA-MB-231) cell lines. Compound 9d showed notable cytotoxic effects against the HT-1080 and MCF-7 cells with IC50 values of 25.77 and 27.89 µM, respectively, while compound 9c displayed significant activity against MCF-7 cells with an IC50 value of 25.03 µM. Density functional calculations at the B3LYP/6-31G* level of theory were used to confirm the high reactivity of the terminal alkyne as a dipolarophile. Quantum calculations were also used to investigate the mechanism of both the uncatalyzed and copper (I)-catalyzed azide–alkyne cycloaddition reaction (CuAAC). The catalyzed reaction gives complete regioselectivity via a stepwise mechanism streamlining experimental observations. The calculated free-energy barriers 4.33 kcal/mol and 29.35 kcal/mol for the 1,4- and 1,5-regioisomers, respectively, explain the marked regioselectivity of the CuAAC reaction.