St. Louis started its comeback when Zach Sanford cut it to 3-1 at 5:16 of the second period. He toe-dragged Foligno before shooting through the forward's legs and over Kahkonen's glove from the left circle.
Schenn scored a power-play goal from the slot to make it 3-2 at 8:29, and Perron tied it 3-3 at 11:33 with a wrist shot from the right circle.
"It was pretty obvious what was going on in the first period," Perron said. "Like [former Blues coach Ken Hitchcock] used to say, 'We just threw our sticks on the ice and played,' and so we had talked about it. [Blues coach Craig Berube] brought it up as well. ... The details, execution, playing the right way in a way respecting the game, doing the right things all the time, I think that matters. We did it in the last two periods. We got rewarded."
Schenn put the Blues in front 4-3 at 14:03 when he scored his second power-play goal on a rebound just outside the crease.
Perron scored with the man-advantage on a one-timer from the point to make it 5-3 at 5:50 of the third period.
Kyrou extended the lead to 6-3 at 6:33 after receiving a pass on a 2-on-1 from Klim Kostin, who got his first NHL assist.