Jordan Binnington made 30 saves in his second straight start, and Justin Faulk and Pavel Buchnevich each had two assists for St. Louis, the No. 3 seed from the Central Division.
"Our team is built to go deep into games and deep into playoffs and deep into the series," Blues coach Craig Berube said. "You know, that's where we're at. Today was a good example. We're down 2-1 and we went out and capitalized and got it done."
Kirill Kaprizov scored two goals, and Kevin Fiala had two assists for Minnesota, the No. 2 seed from the Central. Marc-Andre Fleury made 27 saves.
"The goal isn't for a player to play well," Kaprizov said. "The goal is for the team to play well, and we didn't win. Ultimately, that's what we play for. We have a chance to make some changes to get the series back home for Game 7, and that's what we have to do."
Tarasenko put St. Louis in front 3-2 at 1:03 of the third period, taking a backhand pass from Buchnevich, who was behind the net, and scoring all alone in front.
"Buchnevich] did this many times before," Tarasenko said of the pass. "We talk a lot about staying connected in the O-zone. He's a great skill player and very unselfish all the time. Not surprised he made this play, and that was a great play and big goal for us."
***[RELATED: [Complete Wild vs. Blues series coverage]*
Tarasenko extended the lead 4-2 at 2:31 with a shot glove side from the top of the right circle before scoring into an empty net at 18:27 for the 5-2 final.
Ryan O'Reilly gave the Blues a 1-0 lead at 4:53 of the first period when he converted a rebound at the top of the crease on the power play.
Kaprizov tied it 1-1 at 13:15 with a power-play goal from the top of the left circle through a screen by Marcus Foligno. He then made it 2-1 at 17:07 with another power-play goal, this time shooting short side over Binnington's right shoulder from low in the left circle.
"[Kaprizov is] unbelievable, right? He's unbelievable," Wild coach Dean Evason said. "He's physical. It'd be nice to have 20 of them.
"This kid just continues competing his [butt] off, and goals aside, he wants to win. He wills the team. He just didn't have enough guys willing alongside him tonight."