Clayborn sacked Dak Prescott six times, setting an Atlanta record, and the Falcons snapped the Cowboys' three-game winning streak with a 27-7 victory on Sunday.
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.
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.
Two of the sacks resulted in fumbles and Clayborn recovered one of them.
Prescott, who was 20 of 30 for 176 yards, was sacked eight times.
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 December 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 a 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 had given the Falcons a 10-7 lead at halftime.
Coleman, forced into full-time duty with Devonta Freeman out since the first quarter with a concussion, gained 86 yards on 21 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 year.
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.
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. ... G/T Chaz Green started in place of injured LT Tyron Smith. ... 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.