Jose Ortiz was on board the 5-1 second favourite - trained by Todd Pletcher - to emulate brother Irad Ortiz Jr, who won the same race last year on Creator.
Tapwrit's late charge overhauled Irish War Cry in the closing stages to win by two lengths in New York, with one-eyed horse Patch, a 13-1 shot, in third.
The Triple Crown includes the Kentucky Derby and Preakness Stakes.
It is Pletcher's third victory in the race and his second of the Triple Crown this season, after winning the Derby with Always Dreaming.
Tapwrit won the Belmont Stakes ahead of 3-1 favourite Irish War Cry in the final leg of the US Triple Crown.