Briggs' interception came in a wild two-play exchange of turnovers midway through the third quarter.
Bears quarterback Jay Cutler was nearly flawless after halftime, when he completed 11 of 12 passes for 219 yards and two touchdowns. That included a 34-yard score to Devin Hester to start the half and a 31-yarder to Brandon Marshall with 61/2 minutes remaining.
Cutler was 18 of 24 overall for 275 yards.
Major Wright, who returned an interception for a touchdown in Chicago's last game, had two of the five interceptions. D.J. Moore has nine career interceptions, three against Romo after getting another one Monday night.
Chicago's first fumble of the season came when Cutler was sacked by DeMarcus Ware, and Victor Butler recovered at the Bears' 27.
On the next play, Romo was trying to escape pressure when he was hit from behind by Henry Melton. The ball popped forward into the air and Briggs grabbed it and rumbled for his first interception return for a score since 2005 to put the Bears (3-1) ahead, 24-7.
Once again, the Cowboys (2-2) are a .500 team. They are also one of the lowest-scoring team in the NFL with only 65 points.