Kailer Yamamoto extended the lead to 3-1 lead at 10:36 of the third after redirecting Barrie's point shot past Binnington for his 100th NHL point. Robert Thomas scored 49 seconds later at 11:25 to make it 3-2 on a wrist shot from the right circle.
Ryan Nugent-Hopkins, McDavid and Draisaitl all failed to score on Binnington in the shootout after he made three consecutive saves late in the third to keep it a one-goal game.
"Coming through at the end of the third period like that to tie it up was massive," Binnington said. "You could feel the excitement from the boys on the bench. It's one of those wins we remember moving forward.
"The mindset's just give your team a chance to win right until the end and compete."
Kyrou was the first St. Louis shooter in the shootout and beat Skinner high glove.
"Pick a corner," Kyrou said. "'Buchy' gave me a little advice on a shootout move. That's kind of his move."
NOTES: The short-handed goal was Tarasenko's first in the NHL. ... Thomas has 12 points (five goals, seven assists) in 12 games against Edmonton. ... Buchnevich played 21:57 in his return after missing four games with a lower-body injury. ... Hyman has 15 points (five goals, 10 assists) over his past nine games. ... Barrie has nine assists during a five-game point streak. ... Draisaitl had an assist to give him 20 points (six goals, 14 assists) in 20 games against St. Louis. ... The Blues are the only NHL team McDavid doesn't average a point per game against; he has 18 points (seven goals, 11 assists) in 21 games.