Clayborn entered Sunday with two sacks through the first eight games. He now has a team-leading eight sacks, a 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 took full advantage of Chaz Green, who started at left tackle for the Cowboys in place of injured All-Pro Tyron Smith. All six of Clayborn's sacks were of quarterback Dak Prescott.
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.
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.
Information from ESPN Stats & Information was used in this report.