The Capitals went on a power play in the extra time when Brendan Smith was whistled for tripping Troy Brouwer, and Ovechkin made the Red Wings pay by one-timing a pass from John Carlson past Jimmy Howard for the game-winner.
Carlson notched a goal and an assist and Brouwer also scored for Washington.
Michal Neuvirth stopped 25-of-30 shots to earn the victory.
Gustav Nyqvist logged his first career hat trick and added an assist for the Red Wings, who dropped a third in four games.
Howard finished with 22 saves.