Kyle Okposo and Michael Grabner also scored for the Islanders, and John Tavares, named to Canada's Olympic team earlier Tuesday, added three assists after registering five points against Dallas.
"We got some timely goals, and you take advantage of those when they come your way," Tavares said. "We did a good job trying to limit their chances.
"When you start to win a few, your confidence starts to come back. I still think the ability is here, it's the execution. Obviously, that's been there the last few weeks."
Tyler Bozak and Mason Raymond also scored for Toronto, 14-10-1 at home.
"We're not skating and being the tenacious Toronto Maple Leafs that I expect or everybody expects," Carlyle said. "We don't have enough tenacity in our game right now."
Bozak gave the Maple Leafs a 1-0 lead when he deflected Paul Ranger's point shot past Poulin 5:05 into the game for his sixth goal. Bozak has two goals and four assists in four games since returning from an oblique injury.
Poulin filled in again for Nabokov, who was placed on the injured list Monday night. New York went 1-9-2 when Nabokov was recently sidelined by a groin injury.
Okposo, who sat out Monday because of the birth of his daughter, pulled New York even at 19:16 with his 17th goal.
New York went ahead on a peculiar play early in the second. Bernier left his crease to stop a loose puck behind the net but couldn't prevent it from going outside the no-play zone. That allowed Grabner to take the loose puck and score his sixth goal unassisted on the wraparound at 3:06.
"That was obviously my bad," Bernier said. "I thought the puck was stopped.
"When I looked up to see where the guy was, the puck was still rolling."
Toronto had four power-play chances in the period but only scored at 8:25 to make it 2-2. Poulin stopped Jake Gardiner's point shot but Raymond converted the rebound for his 12th goal and second in 16 games.
NOTES: D Tim Gleason made his Maple Leafs debut. ... Tavares extended his point streak to seven games, and teammate Thomas Vanek earned an assist Nielsen's goal to stretch his string to 10 games.