Pacioretty netted the winner at 7:32 of the second - his team-leading ninth goal. He got the better of Allen on a breakaway after taking a stretch pass from linemate Pierre-Alexandre Parenteau from his own zone.
"That was an amazing pass," said Pacioretty, who added his second of the game in the third period. "I just had a feeling that it was going to land right over. Passes don't get much softer and nicer than that. I credit my first goal to (Parenteau) completely."
The Canadiens (15-5-1) stretched their lead atop the Eastern Conference to three points. The Blues have lost consecutive games after winning 10 of 11.
Montreal coach Michel Therrien was reluctant to give too much importance to the victory.
"It was a great challenge," he said. "This is one of the elite teams in the NHL - really well balanced, and they play really well defensively. It's the kind of game that can give you a lot of confidence.
"But it's a little early to make statements."
Tarasenko scored at 7:45 of the first period, batting the puck out of the air. The goal came on the Blues' third shot.
The period was played largely in the neutral zone and along the boards, and the Canadiens failed to score a goal in the first period for the 16th time this season.
After going 96 minutes, 9 seconds without scoring a goal - dating to the third period against Detroit on Sunday - the Canadiens scored twice in quick succession against the Blues.
Weise jumped on a defensive blunder by Kevin Shattenkirk to tie it 1-1 at 4:50 of the second. Shattenkirk made an ill-advised pass out from behind his net that was intercepted by Weise, who waited for Allen to go down before beating him high for his fourth goal.
"It's a nice bounce for me," Weise said. "Just standing there, throw your stick out front and it hits you. I made a little pump fake, and then saw he took his goalie out. I tried to get in there as quick as I could."
Weise scored only three times in 17 games last season.
Pacioretty scored less than three minutes later.
In the third, after Price made back-to-back sensational saves on two odd-man rushes 20 seconds apart, first on Jori Lehtera and then on Jaden Schwartz, Pacioretty added his second of the game off a pass by David Desharnais.
The Canadiens were well disciplined for the second straight game, taking no penalties against St. Louis' NHL-best road power-play unit (30.4 percent).
"We made some puck errors that ended up in our net," Blues coach Ken Hitchcock said. "We made a big push when it was 2-1 - we got all the scoring chances. Their third one really took the wind out of our sails. We made some casuals plays, and they ended up at the back of our net.
"Montreal has speed, and they really took advantage of our mistakes. They scored on their opportunities, and we missed on ours."
Earlier Thursday, Montreal acquired veteran defenseman Bryan Allen from Anaheim for Rene Bourque.
NOTES: Earlier Thursday, Montreal acquired veteran defenseman Bryan Allen from Anaheim for Rene Bourque. ... Canadiens D Mike Weaver was back in the lineup. ... The Blues came in with a league-low 1.89 goals-against-average. ... St. Louis hadn't lost a game in Montreal since 2004. ... The Canadiens are 12-0 when tied or leading after two periods.