One of the biggest problems facing contemporary medicine is cancer. New approaches to therapy are required due to the difficult and prolonged treatment, the numerous adverse properties of the medications employed, and the developing confrontation of neoplastic cells to treatment. Ten 1,2,4-triazine sulfonamide derivatives (1–10) were chosen for the first time in the current work, and their chemical structures were examined by DFT studies. The in silico flexible docking analysis of the chosen receptors involved in cancer development and metastasis (3RHK, 5GTY, 6PL2, and 7JXH) revealed that the selected compounds are the most promising. The binding affinity of compounds 10, 2, 6, and 4 is much better than the standard drug, Erlotinib, whereas compounds 9, 3, 1, and 7 showed better affinities as compared to standard drugs Neratinib and Tepotinib in the case of 3RHK receptor. The binding affinity against the 5GTY receptor of compounds 10, 5, and 3 is much better than the standard drug Tepotinib, and compounds 7, 6, 2, 4, 1, 8, and 9 showed better than Erlonitib and Neratinib. The binding affinity against the 6PL2 receptor of compounds 8, 3, 5, 4, 9, and 1 is much better than the standard drug Tepotinib. Compounds 10, 6, 7, and 2 were better than Erlotinib and Neratinib. All selected drugs showed better binding affinities than the standard anti-cancer drug Neratinib in the case of the 7JXH receptor, whereas compounds 2, 10, 5, 9, and 8 are better than Erlotinib. In silico ADME experiments supported the identified compounds’ drug similarity. According to the MEP calculations, compounds 3 through 10 can interact non-covalently. The interactions might take the form of σ- and π-hole interactions. Softest compound 4 has the smallest energy gap, with an E-gap value of 3.25 Ev. Compound 4 has the largest energy gap at 3.41 eV. Compound 5 superior electron donor has the highest HOMO energy (6.5470 eV for HOMO). Compound 2 has the lowest LUMO energy, which suggests that it would be the best electron acceptor (ELUMO = 5.766364 eV).