Although the Red Wings suffered a loss, they picked up a point thanks to a game-tying goal by Brian Rafalski with a single second remaining in the third period.
That point kept the Red Wings in a tie for the final playoff spot in the Western Conference. Detroit began the night with a one-point lead over Calgary, but the Flames earned two points with their 4-3 decision over San Jose.
Edmonton, which is in last place in the West, halted a five-game losing streak.
Andrew Cogliano and Ryan Whitney scored in the game's first five minutes, but Edmonton could not register another goal until the shootout.
Devan Dubnyk made 31 saves to get the victory. Jimmy Howard stopped 22-of-24 shots in regulation and overtime, but surrendered two goals on three attempts in the shootout.