Marco Rossi cut it to 2-1 at 1:10. After Blues forward Zack Bolduc fell down while trying to clear the zone, Kaprizov found Rossi all alone in front for a shot five-hole on Binnington.
"They scored to make it 2-0 and it was important that we score," Rossi said. "That gave us some momentum to our side. We kept pushing.
"They outcompeted us in the first two periods. That's not good enough. We had to show a reaction in the third period."
Kaprizov tied it 2-2 at 16:01 when Mats Zuccarello's backdoor pass deflected in off his skate at the right post.
"It was a battle," Fleury said. "We had our best period there in the third. We didn't quit there after that second goal. We kept coming after them.
"It's more fun when we win. I feel like the shootout was my fault. It's 1-on-1 and I feel like I should save everybody. I shouldn't give up two in a row."
NOTES: Kaprizov has scored in four straight games (five goals) and has 13 points (nine goals, four assists) during a seven-game point streak. ... Kyrou has 18 points (eight goals, 10 assists) in 18 games against Minnesota. ... Thomas' assist was his 300th NHL point (81 goals, 219 assists in 381 games). ... St. Louis is 28-3-1 when scoring first this season.