Bouchard grabbed a turnover by Pavel Datsyuk at the blue-line, then lifted a backhand shot over Osgood's glove, giving Minnesota its NHL-best 11th overtime win of the season.
Tied 1-1 with less than four minutes to play in regulation, the teams traded goals just 16 seconds apart. Kris Draper gave the Red Wings their first lead, lifting his own rebound over Manny Fernandez's left leg pad. After the ensuing faceoff, Wild forward Mark Parrish deflected Keith Carney's shot that was heading wide past Osgood.
Fernandez had 29 saves in the win, while Osgood stopped 32 shots in his first overtime loss of the season.
After 43 minutes of scoreless hockey, Minnesota got on the board with a short-handed goal. Mikko Koivu started and finished a two-on-one break, beating Red Wings defenceman Brett Lebda at the blue-line, and then snapping the puck past Osgood for his ninth goal of the season.
Detroit tied the game midway through the third when Holmstrom tipped a Henrik Zetterberg shot through a small gap between Fernandez's blocker and the left post.
It was the second of three games between the Wild and Red Wings in a span of six days. The Red Wings won 3-1 on Friday night in Detroit.
Notes: Osgood was making his first start since Nov. 2. He missed 15 games with a wrist injury. ... Wild winger Stephane Veilluex returned to Minnesota late Saturday evening after spending Friday night in a Detroit hospital. He was struck in the face by a puck and broke his nose. ... Red Wings goaltender Dominik Hasek, who beat the Wild on Friday night, was given a day off to rest and did not make the trip to Minnesota on Saturday. Joey MacDonald was on the bench as the Wings' backup.