"He threw it and I just reached up and tipped it," Williams said. "I was in the right place at the right time."
"That was pretty cowardly," Kubiak said, referring to his turning away from the play. "I've been that way in football through a lot of things. I'll make a call and just walk away. I heard everybody go crazy, so I knew it was good. That was a great sound."
After Carr's 1-yard score put Houston (1-3) ahead 10-6 early in the fourth, Williams got to Culpepper for a 3-yard loss. The sack was the first of his career and his first tackle of the game.
Williams jumped for joy and hugged a teammate as the crowd responded with a standing ovation.
"Now you can't say I don't have a sack," Williams said. "It was a good thing, but at the same time, playing a game and putting the pressure on him, that was a great ending to get a sack."
Johnson finished with nine receptions for 101 yards.
The Texans found a pass rush against the Dolphins, sacking Culpepper on the first play of the game and harassing him all day. Houston had only three sacks entering the game, but got to Culpepper five times. The less-mobile Culpepper has been sacked 21 times this season in his return from reconstructive knee surgery.
He was 23-for-39 for 249 yards and a touchdown.
"We know we're a better offense than we played today for the first three quarters," Culpepper said. "We just waited a little too late to put on a surge."
Kubiak's first victory came against a defense led by former Houston coach Dom Capers, who was fired by Houston in January after a 2-14 season. It broke a five-game losing streak dating back to last season.
"I told our players that when you play teams this time in the season that haven't won a game, they're very dangerous teams," Dolphins coach Nick Saban said. "They're a little bit like where they want to prove something ... and I think their team played awfully well today."
The Dolphins took the lead on a 29-yard field goal by Olindo Mare in the third quarter that was set up by a 19-yard fumble return by Jason Taylor, who sacked Carr and stripped the ball before recovering it.
Houston finally took the lead on a 10-play, 70-yard drive that ended with Carr's run. The defense bit on a fake to Ron Dayne and Carr trotted into the end zone untouched before celebrating by throwing the ball high up into the stands.
Carr completed 22 of 29 passes for 230 yards with an interception and also was sacked five times.