Clayborn sacked Dak Prescott six times, setting a franchise record, and the Falcons snapped the Cowboys' three-game winning streak with a 27-7 victory on Sunday.
"No bigger story today than Adrian Clayborn and what a fine performance that was from him," Atlanta head coach Dan Quinn said. "I'm really pumped for the guys, especially Clayborn. He can say, 'One day I put them down six times.'"
Clayborn, a seven-year veteran, took advantage of the absence of All-Pro Cowboys left tackle Tyron Smith to break the Falcons' sack record set by Claude Humphrey and Chuck Smith.
"I would have never imagined it," Clayborn said. "It's crazy. Our tackles were flushing him out and I was coming off the edge."
Matt Ryan, who was 22 of 29 for 215 yards, threw two short touchdown passes and the Falcons (5-4) won for just the second time in their past six games.
The 32-year-old Ryan, a 10-year veteran, became the fastest NFL passer to reach 40,000 yards, doing it in 151 games.
"I've been fortunate to play with a lot of really good players," Ryan said. "It's a really nice honor to celebrate tonight and then move forward. But, hopefully there will be another 40,000 yards."
Prescott, who was 20 of 30 for 176 yards, was sacked eight times and two of Clayborn's sacks resulted in fumbles, with Clayborn recovering one.
"When we got in those passing situations, we couldn't slow them down," Cowboys head coach Jason Garrett said. "They put heat on the quarterback all day long and, unfortunately, we weren't able to have any efficiency to keep drives alive."
"The defense did a great job, really shutting them down," Ryan said.
Elliott, suspended six games by the NFL for domestic abuse, had a judicial stay rejected on Thursday and can't have another appeal hearing until Dec. 1.
Dallas (5-4) finished with 94 yards rushing. Prescott scrambled for the only Cowboys' touchdown in the first quarter.
The Falcons went 75 yards in 11 plays after the second-half kickoff and scored their first offensive touchdown in the third quarter this season on a 3-yard pass from Ryan to Justin Hardy.
After the Cowboys' Mike Nugent hit the right upright on a 38-yard field goal late in the third quarter, the Falcons went up 24-7 early in the fourth as Ryan hit tight end Austin Hooper from 1 yard out to complete a 72-yard drive in seven plays.
Tevin Coleman's touchdown run from a yard out at the end of an 80-yard drive late in the second quarter gave Atlanta a 10-7 lead at halftime.
Coleman, forced into full-time duty with Devonta Freeman out since the first series with a concussion, gained 83 yards on 20 carries.
Prescott scored on an 11-yard scramble on the fourth play after Ryan's first pass went off the hands of receiver Mohamed Sanu and was intercepted by safety Xavier Woods, who returned it to the 21.
It was the eighth interception against Ryan after he threw just seven last season in his MVP campaign.
Matt Bryant kicked a 50-yard field goal on the last play of the first quarter and added a 29-yarder in the fourth quarter.
The Falcons were 7 of 13 on third-down conversions and scored touchdowns on three of four trips to the red zone while holding Dallas at bay.
"This was just a really good team win, where everyone figured in it on all phases," Quinn said.
Chaz Green, who started the first three games at left guard for the Cowboys, filled in for Smith at left tackle with little success.
"We tried (at times) to chip," Garrett said of giving Green help. "But overall, we didn't protect well enough and there was too much pressure on Dax."
NOTES: Cowboys All-Pro linebacker Sean Lee, who missed two earlier games, didn't return after injuring a hamstring in the first quarter. ... The Cowboys lost S Jeff Heath to a concussion in the third quarter. ... Falcons Pro-Bowl RB Devonta Freeman suffered a concussion on the team's second play from scrimmage. ... Falcons CB Desmond Trufant had a first-quarter interception wiped out by an offside penalty. ... The Cowboys play their next three games at home against Philadelphia, the Los Angeles Chargers and Washington. ... The Falcons play at Seattle next Monday, then have home games against Tampa Bay, Minnesota and NFC South-leader New Orleans.