Nico Strum and Jonas Brodin scored late in the third period to force overtime for Minnesota, which won for the first time in three games since clinching a playoff berth. Fiala, Brodin and Mats Zuccarello each had a goal and an assist.
Minnesota, Boston and Pittsburgh are the only teams without a losing streak longer than two games this season.
In the extra period, Fiala carried into the St. Louis zone, passed to Zuccarello, quickly got it back and beat Binnington.
“Great pass by Zuccy, but equally great shot,” Wild coach Dean Evason said.
Fiala, who had a goal and two assists Thursday, has five goals and six assists in his past seven games.
Sturm scored on a tip in the slot with 5:51 remaining in the third to get the Wild to 3-2, and Brodin scored with 59.1 seconds left in regulation. The Wild twice rallied from two-goal deficits in the third period Thursday before losing in overtime.
“We were obviously frustrated after two periods. It seemed like everything was kind of clogged up out there and the pucks weren’t really bouncing our way. It speaks to the guys in the locker room to just once more get the job done. This is the kind of game that we need to play in two weeks when playoffs roll around,” Sturm said.
Minnesota, 8-1-1 in its past 10 games, remains two points behind Colorado for second place in the West Division. The Wild is now 15 up on the Blues for third. St. Louis leads Arizona by four points for fourth and has three games in hand.
“We sat back too much,” said St. Louis center Brayden Schenn. “When you get a little tentative or sit back looking to block shots and stuff like that, you’re going to give them lanes and they’re taking advantage of it.”
The Blues earned five of six points in the three straight games at Minnesota.
“We’ve just got to turn the page on this, be happy we got results the last several games. We still got a point streak going and we got to keep pushing,” Perron said.
St. Louis scored twice in less than two minutes late in the opening period from about the same spot.
Kyrou beat Kappo Kahkonen from low in the right circle, and 1:49 later, Blais gathered a loose puck below the goal line, skated out uncontested and scored on the short side.
Kyrou has two goals and three assists in the three-game series after contributing just two assists in his prior 14 outings.
Binnington stopped Zach Parise on a breakaway and rebound early in the second period but couldn’t stop a one-timer from Zuccarello a couple minutes later off a blind pass near the corner from Kirill Kaprizov to make it 2-1.
Perron’s rebound goal 6 minutes later pushed the Blues’ lead back to two.
St. Louis was 0 for 1 with the man advantage, ending a streak of nine straight games with a power-play goal. The Blues were 11 for 25 on the power play during that stretch.
St. Louis RW Vladimir Tarasenko (lower-body injury) was on the ice for the morning skate but missed his second straight game. D Torey Krug, second on the team with 28 assists, missed his first game of the season with an upper-body injury. Both are day to day. … Wild C Marcus Johansson missed the game with an upper-body injury but is considered day to day. C Nick Bjugstad played for the first time in 13 games, the last four as a scratch after missing eight contests with an upper-body injury.
Blues: Plays two straight at home against Anaheim starting Monday.
Wild: Host Vegas on Monday in the opener of a two-game series.