The Blues have won 10 of the past 11 meetings between the teams. The Central Division rivals won't see each other again until March 14.
"I dont know if we own them," Backes said. "We can't take anything for granted because they're a force in the Central."
The Blues twice battled back from a one-goal deficit in the third period on Friday, but only had to do it once against the Wild.
St. Louis tied it with 5:33 left in regulation when Patrik Berglund intercepted a clearing pass at the blue line and quickly passed the puck to Backes alone in the right circle. His quick shot beat Backstrom high on the glove side.
"We've got to protect the puck, and I didn't protect the puck and it ended up in the back of the net," Wild defenseman Ryan Suter said.
It was the second St. Louis goal scored off a Wild turnover. Tarasenko gave the Blues a 1-0 lead at 10:20 of the second period off a turnover by Koivu near the right point.
Tarasenko gathered the loose puck and cut across the circles between three players before his wrist shot from the left circle went past a downed Backstrom.
Koivu tied it 1-1 at 14:26.
From a four-player scrum near the right corner, Jason Zucker fed Koivu in the right circle. His quick wrist shot beat Allen high.
St. Louis' Kevin Shattenkirk received a major penalty and was ejected with 4:13 left in the second period for checking Ryan Carter from behind along the boards.
The Wild took advantage when Parise's slap shot from the top of the left circle went through traffic and a screen set by Thomas Vanek before beating Allen with 59 seconds left in the period.
The goal was Parise's 250th in the NHL. He has five goals in seven games since returning from a concussion.
"It was a game we felt like we deserved to win," Parise said. "I thought we played pretty good for the full game and didn't really give them much.
"We spent a lot of good time in their zone, a lot of extended shifts in their zone."
NOTES: The Wild signed D Marco Scandella to a five-year extension through the 2019-20 season. ... Minnesota recalled D Justin Falk from AHL Iowa and reassigned D Matt Dumba. ... The Blues were without C Steve Ott, who pulled a groin muscle on Friday. ... St. Louis placed D Jay Bouwmeester on the injured list and recalled D Petteri Lindbohm from AHL Chicago. Bouwmeester (lower body) missed his fourth game.