Carey Price made 24 saves for Montreal, the No. 4 seed, which has lost three straight, including a 2-1 loss here Monday.
"I think we just need to forget about this one quick," Montreal forward Joel Armia said. "It's not over yet. I feel like we have a good atmosphere in the locker room. Everybody knows that it's not over and we're going to work hard for the next game."
The series winner will play the No. 3 Winnipeg Jets in the second round.
Teams that hold a 3-1 lead are 292-29 (91.0 percent) winning a best-of-7 NHL playoff series.
"We're going to see their absolute best next game and we have to make sure that we're ready to go," Maple Leafs forward Jason Spezza said. "It's important we get some rest here after a tough back-to-back, and then we've got to come ready to play because they're going to be a determined team. The fourth win is the hardest to get."
Alex Kerfoot got the first of his three assists on Nylander's fourth goal of the series at 1:27 of the second period. Galchenyuk took Kerfoot's pass as he drove the slot before making a no-look pass to set up Nylander for a backhand tap-in and a 1-0 lead.
Nylander is the third player in Maple Leafs history to score in each of their first four playoff games, the first since Wendel Clark in 1986. Gordie Drillon scored in Toronto's first five playoff games in 1939.
Spezza made it 2-0 at 12:28 when he deflected Galchenyuk's pass on a 2-on-1. On Monday, Price made a spectacular save, lunging to deflect the puck away on Spezza's shot at an open net.
"He is probably the best goalie of our generation, but I think it's just important that you don't get frustrated when saves are made," Spezza said. "You know saves are going to be made at times, and as a goal-scorer you have to stay around the net. When you're getting chances it's usually a good sign. So I expect him to make saves, you're not going to score every time, but you've just got to stay with it."
Joe Thornton (41 years, 327 days) became the oldest player to score a playoff point for the Maple Leafs when he deflected Spezza's pass inside the left post at 14:56 to make it 3-0.