Clayborn entered Sunday with two sacks through the first eight games of the season. He now has a team-leading eight sacks, one half-sack more than his career high as a rookie with Tampa Bay in 2011.
Clayborn was one short of the NFL record of seven in a game, set by Derrick Thomas in 1990.
Clayborn broke the record previously set by Hall of Famers Claude Humphrey and Chuck Smith. Humphrey had five in a game twice, and Smith recorded a five-sack game against the New Orleans Saints on Oct. 12, 1997.
Green was pulled from the game after the fourth sack he allowed on his side -- all of which were Clayborn's. Green's replacement at left tackle, Byron Bell, gave up Clayborn's sixth sack. Clayborn's other sack was on a Prescott scramble.
"I mean, it's unfortunate," Green said. "You have some good days. You have some bad days. This obviously wasn't my best day. I take this on my shoulders and look at myself in the mirror. Just need to fix the things I need to fix. I'm going to get it fixed."
Prescott was sacked eight times in the game, with Dontari Poe and Brooks Reid also getting to the Cowboys QB. That's the most sacks of a Dallas quarterback since Troy Aikman was sacked 11 times by the Eagles in 1991.
"We're not going to leave somebody out to dry, for the most part," Prescott said. "We're not going to point fingers and say it's Chaz. It's whatever it is. We'll go back and look. I probably could have gotten the ball out faster sometimes. We'll go back and look at it and get better from it."
Information from ESPN Stats & Information was used in this report.