A series of 1,3,5-triazine analogues, incorporating aminobenzene sulfonamide, aminoalcohol/phenol, piperazine, chalcone, or stilbene structural motifs, were evaluated as potential antioxidants. The compounds were prepared by using step-by-step nucleophilic substitution of chlorine atoms in starting 2,4,6-trichloro-1,3,5-triazine. Reactions were catalyzed by Cu(I)-supported on a weakly acidic resin. The radical scavenging activity was determined in terms of %inhibition activity and EC50, using the ABTS method. Trolox and ascorbic acid (ASA) were used as standards. In the lowest concentration 1 × 10−4 M, the %inhibition activity values at 0 min were comparable with both standards at least for 10 compounds. After 60 min, compounds 5, 6, 13, and 25 showed nearly twice %inhibition (73.44–87.09%) in comparison with the standards (Trolox = 41.49%; ASA = 31.07%). Values of EC50 at 60 min (17.16–27.78 μM) were 5 times lower for compounds 5, 6, 13, and 25 than EC50 of both standards (trolox = 178.33 μM; ASA = 147.47 μM). Values of EC50 correlated with %inhibition activity. Based on these results, the presented 1,3,5-triazine analogues have a high potential in the treatment of illnesses caused or related to oxidative stress.