Frederik Andersen made 44 saves for the Maple Leafs (46-28-8), who finished third in the Atlantic Division and will play the Boston Bruins in the Eastern Conference First Round.
Toronto reached 100 points despite losing its final three games (0-2-1). It is the first time the Maple Leafs, who had 105 points in 2017-18, have had consecutive 100-point seasons.
"We haven't had as much to play for here in this last little bit and it's been evident in our play," Maple Leafs coach Mike Babcock said. "And so now we've got to get to the next level and play a good Boston team. We'll be ready."
Charlie Lindgren made 44 saves in his season debut for Montreal, which was eliminated Friday.
Poehling, who scored on each of his three shots, drew the Canadiens within 2-1 at 11:41 of the first period when he redirected Jordan Weal's pass from behind the net while he cut across the goal mouth.
Poehling tied it 2-2 at 2:56 of the second, scoring stick side after his deflection of Jeff Petry's point shot hit the right post.
Weal gave Montreal a 3-2 lead with a power-play goal at 7:04, but Trevor Moore tied it at 3-3 at 8:47. Shaw's 19th goal put Montreal ahead 4-3 at 10:17, but William Nylander tied it at 4-4 at 17:33.
