Jets captain Andrew Ladd scored his 16th and 17th goals of the season, tying the game 2-2 with 5:07 left in the third period for Winnipeg (30-26-7), which had a two-game winning streak end.
Ladd went in alone on goalie Anders Nilsson and beat him with a low shot on the blocker side at 14:53.
In the final three minutes, Nilsson was called for delay of game when he shot the puck over the glass, but the Jets couldn't capitalize on their fifth power play of the game.
Nilsson made 36 saves and Pavelec stopped 24 shots.
The victory came on a day when the Islanders needed a lift.
Islanders winger Thomas Vanek was a healthy scratch amid rumours he'll be traded before Wednesday afternoon's trade deadline. Captain John Tavares is out for the rest of the season with a knee injury he suffered playing for Canada in the Olympic Games.
Winnipeg is now one point back of the final wild-card spot in the Western Conference, which is held by the Dallas Stars, who lost 4-2 Tuesday in Columbus but have a game in hand.
Wnnipeg sits at 67 points, one ahead of Vancouver, which was in Phoenix Tuesday night.
Winnipeg opened the scoring when centre Bryan Little won a faceoff against Frans Nielsen and pushed the puck ahead to Ladd, who went to the front of the net and beat Nilsson between the pads at 10:57.
The Islanders, who were dominating the first period, kept pressuring Pavelec and got a lucky bounce when Ryan Strome's shot went off the backboards and out onto McDonald's stick and he fired the puck in at 17:30 to tie the game.
New York scored 1:33 into the second period on a goal that was reviewed for high sticking.
Lee went in alone on Pavelec, who got a piece of his shot, but the puck went high behind him and Lee put his stick up and knocked it down and into the net for his third goal of the season.
The Jets outshot the Islanders 17-7 in the middle frame to up their margin 26-20, but they lost rookie Mark Scheifele to what appeared to be a knee injury.
The young centre took a hit from defenceman Calvin de Haan with just under nine minutes left in the period, fell to the ice and went straight to the dressing room.
New York got its first power play early in the third period, while Winnipeg had come up empty on four tries at that point. The Islanders couldn't score and the Jets upheld their strong penalty kill, which was fifth in the NHL heading into the game.
Winnipeg hosts Los Angeles on Thursday.
Notes: Tuesday marked Paul Maurice's 1,100th game as a head coach.