With Dallas trailing 20-17 in the final minute of regulation, Tony Romo (35-of-50, 313 yards, one touchdown, no interceptions) drove the Cowboys 66 yards on eight plays, setting up a 32-yard Bailey field goal with 2 seconds remaining.
Romo had also rallied the Cowboys (5-5) from a 13-0 halftime deficit. The win handed Dallas a second straight victory and moved the Cowboys within one game of the New York Giants for first place in the NFC East.
Dez Bryant hauled down 12 passes for 145 yards and a fourth-quarter touchdown.
Brandon Weeden went 20-of-35 through the air for 210 yards with two TD tosses to Benjamin Watson for the Browns (2-8).
Trent Richardson racked up 144 total yards as Cleveland fell for a club-record 12th straight time on the road.
Celebrity Families of 2014 [PHOTOS]