Kelly scored on a rebound of a shot from the left point by defenseman Matt Bartkowski. It was Kelly's second goal of the preseason.
The Bruins outshot the Capitals 38-14. Tuukka Rask made 12 saves for the Bruins, while Braden Holtby stopped 35 shots for Washington.
"We're going to have to make some tough decisions at the end [of training camp]," Bruins coach Claude Julien said.
Connor Carrick scored 11:16 into the second period to give the Capitals a 1-0 lead on their second shot of the night.
"He's impressed us throughout the summer and done a good job," Washington coach Adam Oates said of Carrick. "He played good tonight. It's not always going to be that easy, but it's a great sign. He's playing very well."
Chara scored the game's next two goals, at 18:22 of the second period and 22 seconds into the third period on a 5-on-3 power play. Martin Erat tied the game for Washington at 6:38 of the third.
"He's got the long reach and retrieving those loose pucks, [and] he's very good at it as well," Julien said of Chara. "I think there's a lot more qualities in Zdeno up front than people think. I know people see the size of him for a screen, but he does more things than that to be valuable in front."
Material from wire services was used in this report.