The Rangers successfully killed off a double minor when Mika Zibanejad was called for high sticking, which turned out to be a turning point.
"Special teams are huge," said Derek Stepan, who had an assist on Vesey's first goal. "Each night you've got to go out and win your special teams, whether it be on the penalty kill or the power play. That gives you your best chance to win the game."
Then it was Vesey who scored his pair three minutes apart to turn a 2-1 deficit into a 3-2 advantage.
With the pair, Vesey is the third rookie this season to have a multi-goal game, joining Toronto's Auston Mathews and Winnipeg's Patrik Laine.
"I'm trying to get him to be as comfortable as he can because as he gets more comfortable, he seems to be more confident with the puck," Stepan said. "Tonight, some of the plays he made tonight were at a really high level. You can see his game as he gets more comfortable and as he gets used to his linemates and talking a little bit - gotta get him to open up a little bit - but as he's getting more comfortable and more confident he's going to be a big part of our group."
Nash iced the game with an empty net goal with 5.6 seconds remaining.
After the tough start, Henrik Lundqvist was superb over the final 40 minutes, including his best stop when he denied Ovechkin on the doorstep to preserve the one-goal lead late in the second. He finished with 26 saves.
"I loved the way we win this game. I loved the way we battled back," Lundqvist said. "The way we believe in ourselves and in our system. We played a really strong game. Being down two after the first, it was tough but we all felt we could do it."
The Rangers were without defenseman Dan Girardi, who skated this morning but was unable to play. Forward Pavel Buchnevich remained out of the lineup with back spasms.
The Blueshirts return home Sunday night to host the Arizona Coyotes.