The Capitals found themselves trailing 2-1 late in the third period after New York's Thomas Vanek scored with less than 2 minutes to go.
But, with an extra attacker on the ice, they tied the score with 49 seconds remaining on a goal by Nicklas Backstrom.
Then in the extra frame, Ovechkin fired a shot that Islanders goalie Anders Nilsson stopped but didn't control and Ovechkin was there to knock the loose puck in for the game-winner.
Aaron Volpatti also scored and Braden Holtby made 37 saves as Washington notched a second straight win.
Cal Clutterbuck scored and Nilsson stopped 27 shots for the Islanders.