Brother Irad Ortiz Jr rode Creator to an upset win a year ago.
Tapwrit then made his move down the final stretch, trying to catch Irish War Cry.
The Belmont was already without the Kentucky Derby (Always Dreaming) and Preakness (Cloud Computing) winners.
Tapwrit thrived in the 1½-mile "Test of the Champion" despite never having won beyond 1 1/16 miles.
The 2017 Belmont Stakes was an interesting race from the onset. "It was everything we talked about in the paddock before the race". "At the eighth pole, I thought we might be home free", Irish War Cry trainer Graham Motion said.
That day he bumped into the highly promising Thunder Snow, and was beaten a head by that horse, who has since ran well behind British Guineas double victor Churchill. The colt had been treated for lameness in his right front hoof earlier in the week.
Irish War Cry is another with significant chances after being rested for the Preakness Stakes following an uncharacteristic flat performance in the Kentucky Derby. "I don't think it would be a surprise if either of them ran well".
NEW YORK (AP) - The Belmont Stakes is down to 11 horses after Epicharis was scratched by a veterinarian. "We still have time, so we will give him the best care we can".
So did 86-year-old owner Isabelle de Tomaso, who has her first Belmont starter since she started owning thoroughbreds in 2003. "Tapwrit was getting a attractive trip".
Elsewhere the Acorn Stakes is the first Grade 1 race on the card and in this one Abel Tasman will be very tough to beat if coming here in the same form as she did when she won the Kentucky Oaks. Irish War Cry went for the early lead and was immediately pressured by Meantime, with Tapwrit settling in third. "Big Red", as he was known, became the ninth Triple Crown victor and the first for 25 years. "It's a tough race".
Hollywood Handsome was pulled up after clipping heels with the horse in front of him, causing jockey Florent Geroux to lose his stirrups in the first turn.