Trailing 3-1 in the third period, the Senators staged a comeback that began when Chris Neil beat Niklas Backstrom with a wrist shot 6:13 into the frame. Minnesota's Marek Zidlicky was whistled for high-sticking less than 2 minutes later and Colin Greening made the Wild pay with a power-play marker.
After a scoreless overtime, Matt Cullen kept Minnesota's hopes alive in the shootout with a third-round goal against Craig Anderson, but Alfredsson then deked Backstrom on the final try for the game-winner.
Peter Regin had a pair of assists and Anderson turned aside 24 shots for Ottawa.
Minnesota's Kyle Brodziak, Devin Setoguchi and Nick Johnson scored goals and Backstrom stopped 41-of-44 Senators shots.