Jared Spurgeon and Zach Parise scored, and Alex Stalock made 17 saves for Minnesota (34-30-9), which moved within one point of the Arizona Coyotes for the second wild card into the Stanley Cup Playoffs from the Western Conference.
"The lost point hurts," Parise said. "It's that time of the year where you've got to get them. But at the same time, that just feels probably as well as we've played in a while.
"You go through the game, I don't think there's too many things that we did poorly or we need to do differently. Just they got that one in overtime. Unfortunately, the standings, there's just not that breathing room where we need to get those."
Barzal stripped Wild forward Jordan Greenway of the puck and passed cross-ice to Jordan Eberle, who scored on the Islanders' first shot of the game at 9:28 of the first period for a 1-0 lead.
Wild forward Ryan Donato thought he tied the game at 4:57 of the second period, but Trotz challenged for interference and video review determined goaltender interference occurred to keep it a 1-0 game.
"I'm never confident on any challenge, but I felt pretty good once I saw it," Trotz said. "[Donato's] jamming his stick, but he wasn't making contact with the puck. His stick was right into Greiss' leg, and the puck rolls in just because he pushed the goalie in. So I felt pretty confident, but I leave a lot of that to our video coaches. They looked at it and said, 'let's challenge it,' and we did."
But Parise tied the game with his 25th goal of the season during a power play goal at 11:11 of the second to make it 1-1. It was Parise's first goal since Feb. 21, and the first power-play goal allowed by the Islanders in eight games.
Lee continued his goal streak when he redirected a shot by Devon Toews to end New York's 0-for-25 drought on the power play to make it 2-1 at 16:12 of the second period.
"Tonight, the movement (on the power play) was great," said Barzal, who has six assists in his past six games. "Guys are shooting and guys are coming down on the net. [Lee's] the king of tipping pucks in front, and he had a great one tonight."
Spurgeon tied it 2-2 at 7:15 of the third when he one-timed Eric Staal's pass for a power-play goal.
Each team played the second game of a back to back. New York lost 2-1 at the Detroit Red Wings on Saturday; Minnesota defeated the New York Rangers, 5-2.