Montembeault denied Brendan Smith on a breakaway after he exited the penalty box at 13:35, and later stopped Dawson Mercer on a breakaway at 2:50 of the third period.
"I had a game with two breakaways this year, but three, I'd have to look back," Montembeault said.
Rem Pitlick pushed the lead to 4-1 at 5:13 by scoring at the right post.
Mercer pulled the Devils within 4-2 at 11:58 with a power-play goal on a rebound at the left post to extend his goal streak to four games (five goals).
Matheson scored an empty-net goal at 16:13 for the 5-2 final.
"I think as coaches you're trying to convince them to do things a certain way," St. Louis said, "and when they do it and get rewarded, it's a lot easier to convince them through the success we're having. It's a buy-in and we have that."
NOTES: Montreal defenseman Corey Schueneman had a game-high five blocked shots in 13:20 of ice time. ... Hischier was 16-for-26 on face-offs (62 percent). ... Hughes, who has 71 points (35 goals, 36 assists) in 53 games, became the fastest player to 70 points in a season in Devils/Colorado Rockies/Kansas City Scouts history (John MacLean, 56 games, 1988-89). He has 100 NHL assists and required the fourth-fewest games (219) to reach the mark in franchise history (Scott Gomes, 158; Kirk Muller, 189; Brian Rafalski, 204).