In an effort to improve and achieve biologically active anticancer agents, a novel series of 1,2,3-triazole-containing hybrids were designed and efficiently synthesized via the Cu-catalyzed azide-alkyne cycloaddition (CuAAC) reaction of substituted-arylazides with alkyne-functionalized pyrazole-[1,2,4]-triazole hybrids. The structure geometry of these new clicked 1,2,3-triazoles was explored by density functional theory (DFT) using the B3LYP/6-311++G(d,p) level; also, the potential activity of the compounds for light absorption was simulated by time-dependent DFT calculations (TD-DFT). The antitumor impacts of the newly synthesized compounds were in vitro estimated to be towards the human liver cancer cell line (HepG-2), the human colon cancer cell line (HCT-116), and human breast adenocarcinoma (MCF-7). Among the tested compounds, conjugate 7 was the most potent cytotoxic candidate towards HepG-2, HCT-116, and MCF-7, with IC50 = 12.22, 14.16, and 14.64 µM, respectively, in comparison to that exhibited by the standard drug doxorubicin (IC50 = 11.21, 12.46, and 13.45 µM). Finally, a molecular docking study was conducted within the epidermal growth factor receptor (EGFR) active site to suggest possible binding modes. Hence, it could conceivably be hypothesized that analogies 7, 6, and 5 could be considered as decent lead candidate compounds for anticancer agents.