"It's huge, it usually comes back to haunt you," Carlyle said. "It you don't score on your 5-on-3 at some point in the game usually momentum is going to turn in favor of the opposition.
"They get life from it and it sucks life from you."
Michael Grabner also had two goals for New York, and Mark Streit and Keith Aucoin added goals.
Carl Gunnarsson, Nazem Kadri, Mikhail Grabovski and Matt Frattin scored for Toronto.
Grabner gave New York a 4-3 lead at 3:27 of the third, firing a wrist shot past Ben Scrivens for his second of the season before Aucoin scored his first just 1:08 later to put the Islanders ahead 5-3. That was it for Scrivens, who allowed five goals on 20 shots in his third start of the season.
"I was seeing the puck well and in good position," Scrivens said. "Sometimes those things happen.
"You just have to focus on the process and keep playing games the way I feel I can and should play. It's a journey ... it's the only way you can look at it. It's another learning experience and it's a tough one to swallow but I'll take what I can from it."
Scrivens was replaced by James Reimer, who got the win in Pittsburgh. But Reimer was beaten cleanly by Moulson on a wrist shot at 11:20 for his second.
The Islanders were playing for the first time since beating Tampa Bay 4-3 on Monday night.
NOTES: Frattin had eight goals and seven assists in 58 games for Toronto last season. ... The Islanders are in Boston on Friday night for the first of their seven back-to-back sets this season.