The reaction of 4-azido-quinolin-2(1H)-ones 1a–e with the active methylene compounds pentane-2,4-dione (2a), 1,3-diphenylpropane-1,3-dione (2b), and K2CO3 was investigated in this study. This approach afforded 4-(1,2,3-triazol-1-yl)quinolin-2(1H)-ones 3a–j in high yields and purity. All newly synthesized products’ structures were identified. Compounds 3a–j were tested for antiproliferative activity against a panel of four cancer cell lines. In comparison to the reference erlotinib (GI50 = 33), compounds 3f–j were the most potent derivatives, with GI50 values ranging from 22 nM to 31 nM. The most effective antiproliferative derivatives, 3f–j, were subsequently investigated as possible multi-target inhibitors of EGFR, BRAFV600E, and EGFRT790M. Compound 3h was the most potent inhibitor of the studied molecular targets, with IC50 values of 57 nM, 68 nM, and 9.70 nM, respectively. The apoptotic assay results demonstrated that compounds 3g and 3h function as caspase-3, 8, and Bax activators as well as down-regulators of the antiapoptotic Bcl2, and hence can be classified as apoptotic inducers. Finally, compounds 3g and 3h displayed promising antioxidant activity at 10 µM, with DPPH radical scavenging of 70.6% and 73.5%, respectively, compared to Trolox (77.6%).