WATCH: All Maple Leafs vs. Islanders highlights
"It was tough giving up a late one there, but a good response," Nelson said. "Huge goal by [Ladd] to get us to overtime."
Nikita Soshnikov, Auston Matthews, Mitchell Marner, Zach Hyman and William Nylander, all rookies, scored for Toronto (24-17-10). Andersen made 28 saves.
"I think you're disappointed," Maple Leafs coach Mike Babcock said. "It's always good to get a point on the road. If it was 2-1, I'd probably wouldn't feel like I do right now. When you give up six goals, especially some of the mistakes we made defensively … for me, to give up free ones, just tap-ins by our net, is not good enough. We've got to get that fixed in a hurry."
Strome gave the Islanders a 1-0 lead 1:23 into the game when he put his own rebound in to extend his point streak to four games.
Soshnikov tied it when he took a pass from Nikita Zaitsev and beat Greiss with a slap shot from the slot at 13:27 of the first period.
Matthews put the Maple Leafs in front when his shot from near the top of the crease managed to squeak between Greiss' pads at 14:51 to make it 2-1. It was Matthews' 24th goal.
Kulemin tied it with 1:47 left in the first. Shane Prince passed to Casey Cizikas in the slot, and his wrist shot was stopped by Andersen but Kulemin quickly scored on the rebound to make it 2-2.