Romo broke Aikman's record with his 166th career touchdown pass in the third quarter, when Bryant took a short pass after a scramble by Romo and weaved through the Philadelphia defense for a 23-yard play.
"He made a lot of good throws, different kinds of throws in the ballgame," Dallas coach Jason Garrett said.
Brown's fumble snapped a streak of eight straight scoring drives by both teams. It was the second straight week that he mixed big runs with critical fumbles after losing the ball twice in last week's loss to Carolina.
Foles, making his third start in Michael Vick's place, led the Eagles to a 27-24 lead early in the fourth quarter on a 43-yard field goal by Alex Henery, who now has the longest current field goal streak at 21 after Cleveland's Phil Dawson had a kick blocked Sunday.
Romo had another response, and ended up completing all 10 passes in the second half and his last 12 overall. He was 22 of 27 for 303 yards with no interceptions.
Dallas running back DeMarco Murray, who started after missing six games with a sprained right foot, finished with 83 yards and a touchdown.
Brown, who started his first game since high school when he filled in for LeSean McCoy last week, went in untouched on both of his scoring plays in the first half. He scooted around the left side for a 7-0 lead and trotted through a big hole up the middle to make it 14-3 midway through the second quarter.
Vick and McCoy are sidelined by concussions.
NOTES: The Cowboys snapped an eight-game losing streak on Sunday night. ... Foles' first two career TD passes were against Dallas, and both were to Riley Cooper. ... Heisman Trophy contender Johnny Manziel of Texas A&M watched the game from a suite. His next game will be at Cowboys Stadium, against Oklahoma in the Cotton Bowl on Jan. 4.
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