The Blues scored three times in 3:13 to tie the game.
Brandon Saad's power-play goal at 7:53 made it 5-3, a redirection over Soderblom's left shoulder at the right post.
"I think any time you have good special teams and score on the power play and create some momentum and then we got out there and we got another one," Saad said. "Obviously the power play was huge tonight in coming back."
Jake Neighbours tipped in Faulk's one-timer from the left circle at 10:43 on the power play to make it 5-4, and Kyrou tied it 5-5 at 11:06 from the slot.
"To be honest, I saw 'Buchy' and 'Rouzy' over there and just kind of put it over there," Thomas said of the Kyrou goal. "I got lucky and apparently it landed on his stick and good finish from him."
NOTES: The Blues outshot the Blackhawks 22-2 in the third period. ... St. Louis scored its five third-period goals in a 10-shot span, including consecutive shots on the first three goals. ... Thomas has 10 points (three goals, seven assists) in his past five games. ... Bedard has six points (one goal, five assists) in a four-game point streak. ... Kyrou has eight points (four goals, four assists) in his past five games and had a game-high seven shots (14 attempts) on goal. ... Buchnevich has eight points (three goals, five assists) in his past six games. ... Soderblom is 0-5-1 in his past six starts. ... St. Louis is 12-2-2 in its past 16 games against Chicago dating back to April 3, 2019 and 7-1-1 on home ice the past eight games.