The Wild, backed by 23 saves from Niklas Backstrom, won 2-1 over the Calgary Flames for their first win in five tries away from Xcel Energy Center.
"Sometimes you need good goaltending and tonight we got it," said Koivu. "More importantly, we got the win, especially the first road win for us."
Minnesota coach Mike Yeo had nothing but good things to say about Koivu.
"In a game like that, a guy who is an offensive player and his first thought, his first priority is still defence and making sure nothing gets in the back of our net," Yeo said. "To me that's what a guy like that should be recognized for. He's the kind of guy that you win hockey games with."
With each team having scored once in the shootout, Koivu sped in on Leland Irving and made a sharp move to his forehand to cleanly beat the Calgary goaltender.
Alex Tanguay was then stopped by the pad of Backstrom, sending the Flames to their eighth straight shootout defeat. The result came on the one-year anniversary of the last time Calgary won a shootout.
"Getting blown out in Vancouver and then we respond with an effort that could have been better. It's frustrating," said Calgary's Blair Jones, who shot first in the shootout and missed.
"The crappy thing about it is you don't score every time you go," added Jones, who is 1-for-3 in his career. "I wish I would have scored. I'm pretty angry at myself for not putting it in the back of the net and giving our team a chance to win."
Kyle Brodziak scored in regulation for Minnesota (6-5-1).
Jiri Hudler scored for Calgary (3-4-3), which opened up a three-game homestand with the first in a stretch of six games in 10 days.
Backstrom improved his career mark against Calgary to 14-9-5 in 30 starts. He entered the night with a 1.91 goals-against average and .932 save percentage against the Flames.
"It's never easy. It seems like it's always a one-goal game," said Backstrom. "It's a fun building, a live building, a good crowd. It's fun to play but it's always tough."
Unhappy with his team's efforts after a 5-1 loss to the Vancouver Canucks, Flames coach Bob Hartley completely retooled his lines. Among the changes, struggling captain Jarome Iginla opened the game on a line with Hudler on left wing and centred by recent AHL call-up Ben Street, who was playing his second career game.
"Our fans certainly deserved that win. We battled, but not as consistently as I would have liked," said Hartley, who reunited Iginla with Tanguay and Curtis Glencross by game's end. "It seems that we dominated some parts and they dominated us in some other parts."
Trailing 1-0 after two periods, Calgary tied it 1:28 into the third when Hudler scored on a two-on-one.
The goal came after Brodziak opened the scoring with 1:38 left in the second period, taking advantage of a turnover deep in the Flames' end when Iginla failed to get the puck out.
It was the first goal of the season for the 28-year-old, who was coming off a year in which he recorded career highs for goals (22) and points (44). This year has been a struggle. Brodziak entered the night with no goals, one assist and his minus-6 was better than only Suter's minus-7.
"Finally," said Brodziak, cracking a smile. "Hopefully they'll come more often. It's tough to explain sometimes when it's not going in. It's not something you can dwell on, you just have to focus on playing a strong game."
With Miikka Kiprusoff out with a lower-body injury, Irving's third consecutive start and 10th career start was also his first ever appearance at the Scotiabank Saddledome.
He didn't have long to settle his nerves as he was tested in the opening minute by Koivu, who pounced on Tanguay's giveaway but had his close-in shot turned aside by Irving. The 24-year-old goalie finished the game with 23 saves.
"I take pride in shootouts as a goaltender but it wasn't there tonight," said Irving, who was also beaten by Zach Parise on the first shot.
Notes: The Flames face Dallas on Wednesday. ... The Wild departed immediately for Vancouver where they'll play the Canucks on Tuesday. ... Calgary dropped to 6-21 all-time in shootouts at home. ... Calgary D Cory Sarich drew back in after sitting out the past four games. ... It was the first of five meetings between the two clubs, the first three coming in a 16-day span. ... Iginla has 35 goals against Minnesota in 68 career games. The only other team he has scored that many goals against is Colorado, but his 35 goals against the Avalanche have come in 90 games. ... Calgary played its 10th game of the season, the last NHL team to reach that mark.