Jonathan Bernier allowed three goals on 20 shots and was pulled at 5:05 of the second period for Semyon Varlamov, who allowed three goals on 11 shots.
St. Louis lost 6-2 to the Minnesota Wild on Tuesday.
"It was a team that had each other's backs tonight," Blues coach Mike Yeo said. "... I kind of felt it in the morning and at the rink there [Wednesday] getting ready for today. Some good conversations with some guys.
"I've said it before, we have a lot of character inside our locker room. It was a disappointing game a couple games ago and we were confident that they were going to bounce back."
Bourque tied the game 1-1 at 2:50 of the second period, but the Blues then scored four times, including twice in 48 seconds.
"Just a bad turnover, and for me, [Bourque] gets me, I played with [Bourque] a bunch and he makes a pretty good move and gets it in," Hutton said. "From there, it's 1-1 and it's still really tight and we answer the bell. It's two quick ones and they switch goalies and we get another few and then it's out of reach."
Ivan Barbashev made it 2-1 at 4:17, and Schenn scored his second at 5:05 to make it 3-1. Kyle Brodziak scored from the slot at 7:38 to make it 4-1, and Alex Pietrangelo's one-timer from the left circle made it 5-1 at 15:01.
"We passed up on some shooting opportunities," Bednar said. "I thought early in the second period we did a good job getting some pucks to the net. We hit a post, one sneaks through [Hutton], a couple rebounds laying there, but we didn't come up with enough of those rebounds when we did shoot it."
Schenn gave St. Louis a 1-0 lead at 8:41 of the first period with a power-play goal after he went to the net for a rebound.
Stastny's rebound goal at 6:00 of the third period made it 6-1.
