Nic Dowd and Garnet Hathaway each had a goal and an assist for Washington (14-4-6), which had won eight straight home games against Chicago since a 4-3 overtime loss on Jan. 10, 2006. Vitek Vanecek made 25 saves.
Evgeny Kuznetsov and Daniel Sprong each hit the post in the shootout before Kane scored.
"Just a little bit unlucky," Kuznetsov said. "It's kind of lottery always, right? We knew that Kane [is] going to have a pretty good chance to score, so I had to at least score one, but it didn't work out, so I guess we'll have to practice, get better."
Alex Ovechkin did not score and remained one goal shy of the NHL power-play record and No. 750 in his NHL career.
Jones tied it 3-3 with a wrist shot from the high slot with 8:04 remaining in the third period off a pass from Josiah Slavin, who made his NHL debut.
"I didn't really have much going to the net, so I thought I'd peel back, and as soon as I did, I saw him jumping up in the play and he was wide open. So it was an easy pass to make," the 22-year-old forward said.
DeBrincat gave the Blackhawks a 1-0 lead at 19:00 of the first period when he received a pass from Kane on a 2-on-1.
"I thought we did a great job of kind of taking the crowd out of it in the first, sticking to our game, keeping it low-eventful," Jones said. "That's just giving ourselves a chance. [Fleury] made big saves."