Robin Lehner made 36 saves for New York (20-13-4), which has won three in a row and six of its past seven games.
"You could tell before the game guys were dialed in, guys were ready to go, we knew what was at stake, knew it was a big game against one of the top teams in the League," Barzal said. "If we play like that, this team can be pretty special. Guys are just battling to win; everyone just wants to win here right now, so it's a great atmosphere."
WATCH: [All Islanders vs. Maple Leafs highlights]
Islanders coach Barry Trotz got NHL career win No. 782, tying Al Arbour for fourth on the all-time list. It was the first time the Islanders shut out the Maple Leafs in Toronto.
"We're playing a 200-feet game, we're doing all the right things, we don't take unnecessary chances," Lehner said. "We're forechecking really well and I think we're back checking really well. As long as we play that way, we're going to be successful."