Washington fell behind 3-0 after one period, but rallied behind two goals from Ovechkin and a game-tying tally from Nicklas Backstrom on the power play with 5:50 to play in regulation.
It was the NHL's first overtime game of the season and came on the third night of the campaign.
Michal Neuvirth replaced Braden Holtby in goal after Holtby gave up the three goals in the first period. Neuvirth allowed a lone goal on 28 shots the rest of the way and then stopped the only two attempts he saw in the shootout.
Ovechkin, however, brought Washington within a goal by the end of the middle period by scoring twice in a span of 4:01, one of the goals coming on the power play.
Karri Remo, appearing in an NHL game for the first time in five seasons, gave up four goals on 39 shots during regulation and overtime before letting in both shots in the shootout.