Mitchell Marner's home point streak ended at 20 games (29 points; eight goals, 21 assists), a Maple Leafs record.
Anders Lee scored twice, and Ilya Sorokin made 34 saves for the Islanders (23-21-5), who have lost five straight games (0-3-2) and nine of their past 10 (1-6-3).
"Situational awareness, individual awareness, that's it," New York coach Lane Lambert said. "Three of their goals, penalty kill goal, bad change, [then) a couple guys getting in behind us. Can't happen. It's unacceptable.
"I thought our first period was our best period we've played certainly this month or one of them anyway and for whatever reason, our details and our awareness cost us tonight."
Lee gave the Islanders a 1-0 lead at 19:37 of the first period on a rebound in the goal crease.
New York outshot Toronto 13-6 in the first.
Nylander tied it 1-1 at 5:07 of the second period off a drop pass from Tavares. It was his 25th goal of the season.
"We regrouped well in the locker room, [we] just realized we came out a little flat," Tavares said. "Their urgency was high, and they were winning a lot of the pucks and we had to get our pace up, win some of those key battles, sustain pressure and tilt the ice. Once that got going, we earned some opportunities."
Lee put the Islanders back in front 2-1 at 6:18 on a one-timer from Kyle Palmieri, who returned after missing 27 of the previous 28 games with an upper-body injury.
Tavares tied it 2-2 on a breakaway at 7:53 on the power play. Samsonov sent a stretch pass to Nylander at the offensive blue line, who found Tavares alone.
"I wasn't expecting that [pass from Samsonov] to be honest; I was happy I got back onside because he sent it up there quickly," Tavares said. "He's done that a couple times this year, his recognition and ability to zip it up there is great, so all facets of the game you can tell he's into it."