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.
Swim Week Miami Beach 2014 [PHOTOS]