Vesey snapped a tie with 4:27 left in the second period. He added an empty-netter with 1:13 remaining for his fifth of the season.
The Rangers improved to 6-1-1 in their last eight games after an inconsistent several weeks. They have won three straight at Madison Square Garden after winning only four of their first 14 home games.
''We're in high spirits,'' Vesey said. ''We're going into every game feeling like it will be a win.''
Filip Chytil also scored for New York, and Vincent Trocheck had two assists.
Toronto lost in regulation for the first time since Nov. 11. It went 12-0-3 in its previous 15 games. The Maple Leafs also had recorded a point in nine straight road games, including seven wins.
Toronto failed to score on three power-play chances.
''Sometimes stuff doesn't go your way,'' said Leafs forward Mitch Marner, who had his points streak stopped at 23 games. ''I think something we can definitely look at is our power play. That was pretty bad, it was horrible really. Not creating a whole lot for our team.''
Vesey, who won a job with the Rangers out of training camp, skated down the right wing before flipping a high wrist shot past Toronto goaltender Matt Murray in the second.
Vesey, 29, played three seasons for the Rangers after four years at Harvard before skating for four other teams, including Buffalo, Toronto and Vancouver. He played 68 games for the New Jersey Devils last season.
''I have come full circle and the end of the day, this is where I have wanted to play all along,'' Vesey said about returning to the Rangers.
Michael Bunting had Toronto's lone goal. Murray made 18 stops.
Shesterkin improved to 5-0-1 in his last six starts and 15-4-4 overall this season. He denied Marner at 14:29 of the third, ultimately stopping Marner's franchise-record point streak.
''It's a cool accomplishment to have your name with some great legends,'' said Marner, just the 12th player in NHL history with a point streak of least 23 games. ''It wouldn't have been possible without the guys in this room, so a lot of love goes out to them.''
Chytil opened the scoring for New York, netting his seventh on a power play 3:52 into the game. He rifled a high shot past Murray.
Bunting responded with his seventh at 14:06, tapping a rebound past Shesterkin. Bunting extended his point streak to 10 games.
The brisk pace continued in the second as both goaltenders made sterling saves.
Murray denied a shot from the high slot by Artemi Panarin seven minutes in. Shesterkin stopped Auston Matthews' point-blank shot midway through the period.
Toronto had a flurry of chances during a man advantage late in the second but couldn't solve Shesterkin.
''We just didn't do enough,'' Maple Leafs coach Sheldon Keefe said. ''I thought defensively we were excellent.''
Shesterkin also stopped Toronto's Zach Aston-Reese five minutes into the third, sparking chants of ''Igor! Igor!'' from the Garden crowd, and then smothered a close-in shot by Toronto captain John Tavares at 12:16 of the third.
''We did a good job of tilting the ice for significant portions of the game,? Tavares said. ''We just need to find a way to get the puck a little more to the net and challenge a little more.''
Toronto was coming off three home wins - a 7-0 thrashing of Anaheim on Tuesday after victories over Calgary and Los Angeles.
Following a stretch of one win in six games, the Rangers started their current winning streak against St. Louis on Dec. 5, and then defeated Vegas and Colorado on the road before rallying past the New Jersey Devils in overtime on Monday.
''Good effort by our group,'' Rangers coach Gerard Gallant said. ''The whole group played really well. A team effort from start to finish. Everybody bought in.''
Defenseman Ben Harpur played his first game for the Rangers since he was recalled from the minors on Dec. 3. ... Forward Vitali Kravtsov returned after being a healthy scratch the past three games as the Rangers scratched forward Sammy Blais - who has yet to score in 27 games - and defenseman Libor Hajek. ... Toronto scratched forward Wayne Simmonds and defenseman Jordie Benn.
Rangers: Visit the Philadelphia Flyers on Saturday.
Maple Leafs: Visit the Washington Capitals on Saturday.
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