Ivan Barbashev scored at 13:03 of the first period to give the Blues a 1-0 lead on a goalmouth scramble.
Hartman tied it 1-1 at 1:41 of the second period, collecting a loose puck and beating Husso from the low slot.
Tarasenko made it 2-1 at 3:24, one-timing a Parayko backhand pass through the slot at the right side of the post. It was his seventh goal in the past four games.
David Perron's one-timer from the left circle at 6:52 made it 3-1 as a power play expired.
Brandon Saad made it 4-1 at 13:38 with a backhand.
"Our puck play in the third period wasn't very good," Blues coach Craig Berube said. "We turned some pucks over. They capitalized on them, tied it up, got the lead again, kind of a back and forth game in the third. Not like we wanted it, but we hung in there and got a win."
NOTES:Kaprizov had been tied with Marian Gaborik (2007-08) and Eric Staal (2017-18) for the Wild single-season goals record. ... Minnesota forward Mats Zuccarello (two assists) set the team record for assists in a season with 52. ... St. Louis is 12-1-1 against Minnesota the past 12 games, including winning seven in a row. … The Blues have scored four or more goals in 11 straight games, a St. Louis record. ... Blues center Robert Thomas had an assist to extend his NHL career-high point streak to 12 games (24 points; six goals, 18 assists). ... Buchnevich extended his point streak to nine games (17 points; six goals, 11 assists). ... St. Louis defenseman Nick Leddy left the game in the second period after taking a high stick from Wild forward Kevin Fiala. Berube said it's not likely Leddy plays Sunday at the Nashville Predators. ... Minnesota forward Tyson Jost sustained a lower-body injury in the second period and will not play Sunday against the San Jose Sharks, Evason said.