WATCH: All Maple Leafs vs. Predators highlights
The Maple Leafs and Senators each have six games remaining, one more than the Bruins.
James van Riemsdyk, Auston Matthews and Connor Brown scored, and Frederik Andersen made 29 saves for the Maple Leafs, who have won two games in a row and five of six.
The Predators (39-27-11) remain in the second wild card into the Stanley Cup Playoffs from the Western Conference, 10 points ahead of the Los Angeles Kings. They trail the St. Louis Blues by one point for third place in the Central Division.
Van Riemsdyk scored on the power play to give the Maple Leafs a 1-0 lead with 1:18 left in the first period.
Mitchell Marner's assist on van Riemsdyk's goal was his 41st, setting a Maple Leafs record for most by a rookie. The previous record was set by Gus Bodnar in 1943-44.
"It's pretty cool," Marner said. "Obviously, it's nice to do it in a win. I didn't really do much, kind of got it up to [Zaitsev] and he got the shot through. It's definitely pretty cool, growing up in that city and watching the team for my whole life."
Matthews gave the Maple Leafs a 2-0 lead when he scored on the power play with 1:17 left in the second period. He took a pass from Brown and beat Rinne for his 36th goal, most by an NHL rookie this season.
"These games have kind of been a struggle for us at the beginning of the year, especially against good teams," Matthews said. "At this point of the season, being able to close out a close game like tonight is definitely a positive sign for us and something we'll take moving forward."
Filip Forsberg scored to get the Predators within 2-1 at 9:37 of the third period on a wrist shot from the right faceoff circle. He took a drop pass from Ryan Johansen and beat Andersen for his 30th goal.
"We were in one-goal reach and obviously played better in the third," Forsberg said. "We got a couple of chances that we just [couldn't] finish off, and it's frustrating for sure."
Brown scored into an empty net with 59 seconds remaining to make it 3-1.
Andersen returned after missing one game with a jaw injury.