Alexander Radulov gave Dallas a 3-2 lead at 15:53 of the third period, but Brett Connolly tied it 3-3 41 seconds later with an assist from Burakovsky.
"It was really deflating," Stars defenseman Dan Hamhuis said. "We were obviously quite pumped up after [Radulov] scored to go ahead that late in the game. That's where you really want to close those games out, ensure that you get the two points. It was a big blow for sure."
Burakovsky gave Washington a 1-0 lead at 10:35 of the first period on a wrist shot. Connolly set up the goal with a backhand pass on the rush before Burakovsky scored his second goal of the season, his first since Oct. 20. He returned Dec. 8 from a thumb injury.
"It was nice to see Burky back on the goal-scoring sort of chart," Capitals coach Barry Trotz said. "I'd say he was probably pressing a little bit the last couple games, so it was good to get him a couple. I thought we just got contributions from the whole lineup, which was good."