Scottie Upshall also scored for the Blues (18-8-2) and Jake Allen made 22 saves.
Shea Weber scored two goals, including the 500th point of his career, and Jordie Benn scored for the Canadiens (13-13-3). Carey Price made 26 saves. It was Price's first loss since returning from a lower body injury six games ago.
Schenn's third goal at 12:11 of the third period, which deflected off the skate of Montreal defenseman David Schlemko, broke a 3-3 tie and gave the Blues the win.
After the Blues scored goals seven seconds apart by Upshall and Schenn early in the second period for a 3-1 lead, Weber cut it to 3-2 with 2:56 left in the second period when he scored off a faceoff won by center Andrew Shaw. His shot sent the puck rolling on its edge, starting outside the right post and curling by Allen's skate.
Weber tied it at 6:34 of the third period on a similar play: a Shaw faceoff win in the right wing circle back to Weber, who scored on a slap shot through Allen's pads.
The Blues sliced through the Montreal defense to take the 3-1 lead early in the second period.
First it was Dmitrij Jaskin skating down the right wing against Benn and finding room to slip a backhand pass across the slot to Upshall, who beat the check of Montreal wing Brendan Gallagher to get a shot off and score his fourth goal of the season.
Schenn scored his second goal of the game when he took a pass from Alex Pietrangelo in the neutral zone, split the Montreal defense and scored to the glove side on Price.
Benn had tied the game 1-1 at 15:28 of the first period when his rising shot from the right point went in over Allen's right shoulder with Canadiens forward Daniel Carr providing a screen. It was Benn who drew the penalty that led to the power play when he was hooked by Blues left wing Alexander Steen.
Schenn ended a five-game streak without a point when he made it 1-0 at the five-minute mark, snapping home a pass from behind the net by Jaden Schwartz after Schwartz had chased down Schenn's rebound at the right post.
NOTES: With his 500th point, Canadiens D Shea Weber became the fourth active defenseman to reach that plateau, joining Zdeno Chara of the Boston Bruins, Duncan Keith of the Chicago Blackhawks and Brent Burns of the San Jose Sharks. ... Scratched for the Blues were RW Vince Dunn, C Oskar Sundqvist and RW Chris Thorburn. Dunn was scratched for the second game in a row after playing in the Blues first 26 games. ... The Blues' injuries are long term. LW Zach Sanford is out four to five months with a shoulder injury and C Robby Fabbri is out for the season with a knee injury. ... Canadiens C Jonathan Drouin missed his third game with a lower body injury. He is day-to-day. ... Also injured for the Canadiens were RW Ales Hemsky (concussion), LW Artturi Lehkonen (lower body), G Al Montoya (concussion) and RW Nikita Scherbak (knee).