Wes Walz broke loose on the right side and fed Laaksonen, who got his 10th goal of the season and first in nine games since scoring against Anaheim on New Year's Eve.
Giguere was seeking his second straight shutout of Minnesota but gave up a goal to defenseman Nick Schultz with 10:33 remaining in regulation that tied it at 1-1.
Pascal Dupuis worked the puck behind the net and got it to Schultz, who beat Giguere with a quick, low shot. Both of Schultz's goals this season have come against Anaheim.
It looked like Petr Sykora's goal 2:53 into the game was going to hold up for Giguere's third 1-0 blanking of the Wild this season. Sykora tapped in the puck from the left side of the net after defenseman Fredrik Olausson faked a shot from the right point.
Minnesota improved 6-1-7 in overtime and 10-6-4-1 on the road as it opened a five-game trip. The Wild do not return home until after the All-Star break.
Dwayne Roloson made 30 saves for his 13th win.