Alexey Toropchenko put St. Louis up 2-1 at 6:59 when he drove the slot and shot past Allen stick side.
Kirby Dach tied it 2-2 with a power-play goal at 16:15 of the second period, scoring from the slot off a pass from Nick Suzuki.
Cole Caufield gave Montreal a 3-2 lead 36 seconds into the third when he lifted a backhand over Binnington's right arm after Suzuki forced a turnover behind the net.
Alexandrov tied it at 3-3 at 2:29, scoring on a rebound at the right side of the net in his 12th NHL game, and Pavel Buchnevich's power-play goal put St. Louis ahead 4-3 at 4:42.
Armia tied it 4-4 at 7:20 with his second goal of the game. He scored stick side on a shot from the left side for his third goal in two games after failing to score in his first 26. He was a healthy scratch for a 6-3 loss at the Nashville Predators on Tuesday.
"I just made sure I'm ready to go whenever I played next," Armia said. "That was probably the main mindset there, just to get yourself ready mentally and physically to play whenever you're called up."
Blues defenseman Robert Bortuzzo left with a lower-body injury 12:01 into the first period and did not return. Berube had no further update.
"It's not the ideal situation, but we've all been through it before," Faulk said. "It's something that happens every now and then and you've got to find ways to play a good hockey game. I think the game was obviously there for us tonight and we just didn't execute well enough, didn't defend well enough to get the result we needed."
NOTES: Caufield's goal was his 50th in 117 regular-season games. He is the fourth-fastest Montreal player to reach the milestone, among those to have debuted for the Canadiens since 1942-43. Maurice Richard scored his 50th in his 76th game. Jean Beliveau (105 games) and Bernie Geoffrion (114) are the only other Canadiens players to reach the mark faster than Caufield. … Allen is 2-1-0 in three starts against the Blues after playing his first seven NHL seasons (2012-20) for them. He won the Stanley Cup with St. Louis in 2019 and ranks second in its history with 148 wins (Mike Liut, 151) and 21 shutouts (Brian Elliott, 25). … Canadiens defenseman David Savard had two shots on goal and played 22:36 in his 700th NHL game. He and Armia each had three of Montreal's 13 blocked shots. … Suzuki, Evgenii Dadonov and Jake Evans each had two assists for Montreal. … Faulk played 25:54 in his 800th NHL game.