After Alexander Semin put Washington ahead 1-0 in the shootout and Patrice Bergeron evened it, Kessel flipped a backhander past Olie Kolzig to give Boston its second consecutive victory following a five-game losing streak.
Ovechkin and Eric Fehr missed for Washington in the shootout; Brandon Bochenski missed for Boston.
Boston led 1-0 heading into the final period, thanks to 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, and Eric Fehr put the hosts ahead 2-1 at the 7:48 mark.
Boston tied it 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.
The win allowed Boston to pull even with Washington in the Eastern Conference standings - with 52 points apiece, the teams are on the outer fringes of the playoff race.
Ovechkin is mired in the first mini-slump of his NHL career. During his Calder Trophy-winning rookie season of 2005-06, the Russian never went more than two consecutive games without an assist or goal - and he only had three droughts of that length, each time snapping it with a multiple-point game.
Ovechkin, who entered Tuesday tied for the league lead with 33 goals, hadn't even been held without a point in two consecutive games this season until the current stretch.
He only managed one shot on goal through the second period, which ended with a chorus of scattered boos. To that point, the Capitals had outshot the Bruins 20-10, but had nothing to show for it.
That changed quickly. Laich took the puck away from Boston's Shean Donovan and sent a shot past Tim Thomas to even the score.
Washington went ahead when rookie defenceman Mike Green dived to keep the puck from leaving the offensive zone, but it went to Richard Zednik, whose shot was blocked. Fehr then poked in the rebound for his second career goal.
But Bergeron tied it, and the game went to overtime, then the shootout.
Boston led 5:06 into the second period. Marc Savard won a faceoff and the puck went to Bochenski, who turned and slid a shot past Kolzig.
It was Bochenski's first game for the Bruins since they acquired him from the Chicago Blackhawks for a minor leaguer Saturday. The right wing spent most of this season in the minors, though he did score two goals in 10 games with Chicago.
The Bruins were without their leading goal scorer, Glen Murray, who was scratched because of a groin injury. He has a five-game point streak, including two goals in the team's previous game that raised his season total to 26.
Notes: Savard had two assists. ... Capitals coach Glen Hanlon (a goalie) and Bruins coach Dave Lewis (a defenceman) were teammates on the Detroit Red Wings. "He was a great communicator. Real smart," Hanlon said. ... The Bruins recalled LW Jeremy Reich from Providence of the AHL before the game.