Galchenyuk, who missed the past two games because of the flu, had his first three-point game since his hat trick Dec. 16 against the Carolina Hurricanes.
Carey Price made 27 saves in his ninth consecutive road victory, passing Rogie Vachon for the Canadiens record.
"It's pretty neat," Price said. "I think that speaks volumes of the character of our team and the way that we compete on the road. We're a very confident team when we go into an opposing team's building and we wind up playing the right way, and we've been rewarded for it."
Brendan Gallagher scored twice, and Michael Bournival had a goal for the Canadiens (39-16-5, 83 points), who won in St. Louis in regulation for the first time since March 10, 2007. P.K. Subban and Andrei Markov each had two assists.
The Canadiens are one point ahead of the New York Islanders for the top spot in the Eastern Conference with two games in hand.
"He's phenomenal," Montreal coach Michel Therrien said. "We play with a lot of confidence when he's there. He's a special player. There's not many special players in the NHL, but he's one of them."
The Blues are 4-5-0 in the past nine games, and the comments from coach Ken Hitchcock reflected the way they have been playing.
"What we're doing is not paying any respect to checking," Hitchcock said. "We're not paying any respect to defense, to managing the puck, to managing the proper way to playing. I don't care what the shots on goal are. When you give up as many odd-man rushes as we gave up in the last two games, we're showing no respect for what matters in the National Hockey League at this time.
"And in the offensive zone, the sense of urgency that we're not playing with, that we've played with all year, is not there. That's why we don't score, that's why we don't get second and third chances, that's why we don't win the front-of-the-net battles. Those combinations are lethal the wrong way."
Galchenyuk gave the Canadiens a 1-0 lead when he deflected Subban's shot from the right point past Allen with 6:38 remaining in the first period.
The Blues, who have been outscored 11-3 in the first period over their past 11 games, have allowed the first goal in eight straight home games. They failed to score in the first period in seven of those games and have gone nine straight home games without a lead after one period.
Galchenyuk put Montreal ahead 2-0 when he took Tomas Plekanec's faceoff win and beat Allen on the short side with a wrist shot from a sharp angle 3:11 into the second. It was his 18th goal.
"The first couple of shifts, I knew it was going to be kind of tough for me to get out there and get my wind," Galchenyuk said. "You obviously want to score as much as you can, and it's a nice feeling, but you try not to think about that too much."
Backes cut Montreal's lead in half when he took Vladimir Tarasenko's pass in the slot and redirected the puck past Price 6:01 into the second for his 20th goal.
But the Canadiens scored two goals in 49 seconds to go up 4-1.
Gallagher made it 3-1 when he took a snap shot that beat Allen with 4:11 left in the second off a 2-on-1 after Tarasenko's blind drop pass in the offensive zone was behind Jay Bouwmeester.
Bournival. who was serving a penalty for roughing, came out of the box, took a pass from Galchenyuk and beat Allen upstairs on a breakaway with 3:22 left in the second.
"You have no control over the hockey game because of the scoring chances you give up off these odd-man rushes," Hitchcock said. "You work your way back in the game like we did today and give up the chances on casual puck play that we're giving up ... we're a team that's made a very good ... a lot great inroads on playing a certain way, and now we don't want to play that way, and we're not interested in playing the way that's been successful here. We want to play a different way right now, and it's really, really hurting us."
Oshie's shot from the blue line got through traffic and past Price at 17:45 of the second after hitting a Canadiens defenseman out high. Oshie's 17th goal made it 4-2.
Gallagher scored his second of the game, 17th of the season, on Montreal's second power play of the game at 15:47 of the third period to make it 5-2.
"I thought we played a great two-way game tonight and got rewarded," Price said.