Capitals star Alex Ovechkin was held pointless for a career-worst third consecutive game.
After Alexander Semin put Washington ahead 1-0 in the shootout and Patrice Bergeron evened it, Kessel flipped a backhand shot past Olie Kolzig to give Boston its second consecutive victory following a five-game losing streak.
Boston led 1-0 heading into the final period, thanks to Brandon Bochenski's goal in his Bruins debut. But Washington's Brooks Laich scored an unassisted goal only 21 seconds into the third to tie it before Eric Fehr put the Capitals ahead 2-1 at 7:48.
Boston tied it at 2-2 on a power play with 6:45 left in regulation on Patrice Bergeron's 17th goal of the season. It came with Ovechkin in the penalty box after being whistled for delay of game.
Washington fell to 1-6 in shootouts this season, while Boston improved to 6-3.