Braden Holtby made 31 saves, and Brett Connolly and Alex Ovechkin scored for the Capitals.
"I thought we needed a little bit more from all our players tonight," Washington coach Todd Reirden said. "It's self-inflicted. We didn't do a great job of managing the puck to give ourselves a chance to get in on the forecheck as much as we did in Game 5."
Staal gave Carolina a 3-2 lead at 3:51 of the third period. Capitals defenseman Matt Niskanen tried to clear the rebound of Justin Faulk's shot, but Hurricanes forward Brock McGinn blocked the clearing attempt and Staal jammed in the loose puck at the top of the crease.
"I was hoping to get one. So was everyone else in the room," Staal said. "I was fortunate enough that it was me. Everyone was a piece of that game. We're going to have to have that in Game 7."
Ovechkin appeared to tie it 3-3 at 10:34, but the play was ruled no goal for goaltender interference. Reirden challenged the ruling, but the call was upheld after a video review.
"For us, we thought the puck was loose," Reirden said. "It was still a puck in play."
Williams scored his first of the Stanley Cup Playoffs to make it 4-2 at 11:58 when he redirected Brett Pesce's point shot five-hole on Holtby.
"I don't expect anything less from [Williams]," Pesce said. "He's such a talent. You know he's going to come up clutch. It's good to know he's on our side and he's our leader."
Hamilton scored into an empty net with 3:06 remaining to extend the lead to 5-2.
"I thought when they scored, when they were up 4-2, that's when the urgency started for us, but that was too late," Capitals forward Nicklas Backstrom said. "Best out of one game. We just got to regroup and look ahead here and make sure we can come out ready."
Connolly gave Washington a 1-0 lead at 5:06 of the first period. He took a backhand pass from Lars Eller behind the net before quickly moving out in front and shooting high to the far post.
