The Coyotes trailed 3-1 coming into the final period but Laurie Korpikoski cut the deficit with 3 1/2 minutes left and Shane Doan added the second of his two goals with 1:46 remaining.
After the scoreless overtime, the game moved to a shootout, where Vermette scored on Michal Neuvirth on the first shot.
Washington's Mikhail Grabovski missed on his ensuing chance and Boedker followed with backhander for a score.
Alex Ovechkin couldn't score on his final attempt, handing Phoenix its sixth win in seven games.
Mike Smith made 29 stops for the Coyotes.
Neuvirth finished with 32 saves.