Tyler Ennis scored the Sabres' goal just 10 seconds into the game, as Buffalo lost at home after consecutive home wins last week.
The Blues' league-best power play stayed productive, scoring in their fourth straight game. It's a stretch that's seen the Blues go 6 of 17 with the man advantage.
St. Louis is 8-1 in its past nine trips to Buffalo. The Blues improved to 7-2 against Eastern Conference teams, outscoring opponents 31-17.
Though the Blues outshot the Sabres 31-24, they got off to a slow start.
After a dump into the Blues zone off the opening faceoff, Ville Leino stripped the puck from Jay Bouwmeester below the left faceoff circle and found Ennis, who beat Halak with a low shot to give the Sabres an early lead.
It took less than a minute for the Blues to tie it, as Morrow pounced on a rebound and beat Ryan Miller at 1:09 of the first.
Tarasenko scored a power-play goal with 36 seconds to play in the first period on a pass from T.J. Oshie, who collected his own blocked shot.
Buffalo has been outscored 31-4 in the first period this season.
It was Morrow's first multi-goal game since April 17, when he scored twice against Montreal. Stewart had two assists.
Paajarvi added his first goal for the Blues when he beat Miller below his blocker with just over six minutes played in the third.
St. Louis' top-ranked road penalty kill stopped both of Buffalo's power plays.
Defenseman Mike Weber returned to the Sabres' lineup after missing 10 games with a hand injury suffered Oct. 25 against Florida.
NOTES: Blues C Patrik Berglund missed his second straight game with an upper-body injury. ... The Sabres sent D Rasmus Ristolainen, LW Johan Larsson and C Mikhail Grigorenko to their American Hockey League affiliate in Rochester on Tuesday. ... Buffalo also sent D Nikita Zadorov to the London Knights of the Ontario Hockey League.