Minnesota's second shooter, Koivu skated down the slot, went left and cut back to his right before lifting a backhander past Jean-Sebastien Giguere. Manny Fernandez stopped Teemu Selanne on Anaheim's third attempt to secure the win.
Ryan Shannon and Ryan Getzlaf also missed for the Ducks, and Pavol Demitra was unsuccessful for the Wild.
It was the third shootout loss for Anaheim (7-0-3). The Wild are 5-0 in overtime and 9-1-0 overall, with the lone loss a 2-1 decision to Anaheim on Oct. 20.
Brian Rolston and Nick Schultz scored goals for Minnesota, which has won all seven home games this season, and 10 straight overall. Andy McDonald and Chris Kunitz scored for Anaheim in its first loss in five games.
Off to the best starts in their team histories, the Wild and Ducks entered the game tied with Dallas for the top spot in the conference with 16 points. Dallas lost Friday night, falling 4-3 at home to Detroit.
With the Wild down 2-1 at the start the third period, Kunitz it at 3:53 when he knocked down a clearing attempt and fired a slap shot from the right circle through traffic and past Fernandez.
Fernandez, who entered the game with an NHL-best 1.53 goals-against average, finished with 32 saves. Giguere, fourth in GAA at 1.81, stopped 31 shots for Anaheim.
Minnesota, which has allowed 17 goals in 10 games, held the Ducks scoreless in four power-play chances. The Wild have killed off 32 straight penalties.
Rolston put Minnesota up 2-1 late in the second period with a tremendous individual effort. Finishing a penalty kill, he carried the puck into the Anaheim zone and dumped it into the right corner. Following the puck, he checked Getzlaf into the boards before heading toward the bench. As Getzlaf circled behind his net, Koivu stripped the puck from him and centred it to Rolston, who fired a low shot past Giguere.
McDonald redirected a pass from Dustin Penner past Fernandez 6:28 into the game to give Anaheim a 1-0 lead.
Minnesota tied it at 15:10 when Nick Schultz tipped a pass from Todd White behind Giguere. It was Schultz's first goal in 28 games.
Notes: Rolston has a seven-game scoring streak. ... The game was the first of three straight road games for Anaheim. The Ducks will plays Chicago on Saturday night and St. Louis on Monday night. ... Anaheim's Todd Fedoruk went to the locker room after Derek Boogaard dropped him with a punch to the face during a second-period fight. Fedoruk did not return. ... Demitra, who has scored at least 20 goals eight times in his career, has not scored in eight games and has one goal this season.