The Islanders hold the first wild card into the Stanley Cup Playoffs from the Eastern Conference, one point ahead of the Pittsburgh Penguins, who have three games in hand.
"We're going to be playing a lot of these type of tight games," forward Zach Parise said.
Dylan Larkin scored for Detroit (28-25-9), which has lost five straight (0-4-1) and has been outscored 24-8 during that stretch. Magnus Hellberg made 32 saves.
"Whether the playoffs [are] in reach or not, every chance you get to play, you want to play your best hockey," Hellberg said. "That's what I'm going to try to keep doing every game."
The Red Wings trail the Penguins, who have one game in hand, by six points for the second wild card.
"I think this is probably crazy to say, but we're still hanging around," Larkin said. "We're still right there. We have to win hockey games."
Parise tied it 1-1 at 2:11 into the third period. Palmieri's shot from the top of the right circle deflected off Sebastian Aho's stick right to Parise, who snapped a shot past Hellberg.
"I heard [Noah Dobson] yelling at me on the way back to skate it and kind of saw they were going to change, and their middle guy was just standing still, so I should beat him most times," Aho said. "Pumped it up to [Palmieri] and he shot it, good rebound and [Parise] buried it."
Lee gave the Islanders a 2-1 lead at 4:39, tipping Hudson Fasching's shot.