Jason Zucker and Nino Niederreiter scored in regulation to help Minnesota improve to 6-1-1 in its past eight. Alex Stalock made 28 saves.
J.T. Compher and Blake Comeau scored for Colorado, and Jonathan Bernier stopped 25 shots.
Coyle scored beat Bernier high in the second round of the shootout, and Stewart went low in the third round. Stalock stopped Nathan MacKinnon and Mikko Rantanen.
Minnesota had all six shots in overtime, including two outstanding chances, but Bernier stopped Eric Staal alone in front and sprawled to stop Zucker with his right pad.
Colorado controlled much of the third period, including a 10-3 edge in shots, but was 0 for 2 on the power play and Stalock stopped Compher on a breakaway and a rebound.
Niederreiter tied it at 2 early in the second with a power-play goal in his fifth straight game to extend his goal streak to a franchise-record tying six games. Brian Rolston also scored in six straight from Jan. 21-Feb. 5, 2008.
On a career-best eight-game point streak, Niederreiter has 12 points in 13 games since returning from a high left ankle sprain Oct. 31.
With a bit of haze still in the air from pregame activity, Zucker's wrist shot beat Bernier 1:13 in for his ninth goal in nine games.
Colorado, which entered the game with the league's worst penalty kill on the road at 68.6 percent, tied it short-handed nine minutes later when Compher connected on a breakaway. Two of Compher's three goals this season are with the Avalanche a man down.
Comeau scored late in the opening frame for a 2-1 Colorado lead.
NOTES: This is the first of four meetings between Central Division rivals, the first season the teams haven't met at least five times. ... Wild RW Mikael Granlund had an assist and has 10 points in a six-game point streak. ... Colorado G Semyon Varlamov (illness) practiced Thursday, but didn't travel with the team. Andrew Hammond backed up Bernier.
Avalanche: Colorado opens a five-game homestand Saturday night against Calgary.
Wild: Minnesota starts a two-game trip in St. Louis on Saturday night.
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ST. PAUL, Minn. (AP) - Charlie Coyle and Chris Stewart scored in the shootout to give the Minnesota Wild a 3-2 victory over the Colorado Avalanche on Friday.