The Raptors, meanwhile, were without Leonard, who helped Toronto start 2-0 by averaging 27.5 points and 11 rebounds in his first two games with the club after arriving in a trade with the San Antonio Spurs.
Said VanVleet: "At that point, you just want to get a shot up and get an attempt, and I was able to knock it down".
Tatum also had 14 rebounds for the Celtics, who have won two of three to open the season.
The last-minute loss was the second in as many nights for NY, which fell to the Brooklyn Nets, 107-105, on Friday, when Hardaway missed the potential game-winning 3-pointer as time expired. The team lost to the Boston Celtics, 103-101 at home, and rookie power forward Kevin Knox for the foreseeable future.
On Saturday afternoon, the news that various stars will sit for their night games started trickling out. That put Toronto back ahead 109-101 with 3 minutes remaining.
Celtics: Aron Baynes got the start with Hayward out and scored 10 points. He is now restricted to playing 25 minutes or fewer.
Knicks: Coach David Fizdale said G Courtney Lee would see specialists Monday after experiencing neck spasms that went down into his chest. Knox stayed on the court briefly, appearing in pain, then was carried off by a pair of teammates. The Knicks said X-rays for Knox are negative.
With Knox immediately sent to the locker room, he was unable to shoot his free throws, thus disqualifying him of any chance to return to Saturday's game.
Still, it wasn't all good for Hayward, who suffered a gruesome left ankle injury five minutes into last year's season opener and missed the rest of 2017-18.
Celtics coach Brad Stevens said there wasn't a plan to rest Hayward on back-to-backs, but his minutes are being watched and he's being evaluated on a game-by-game basis.