Carr picked off Ben Roethlisberger on the second play of overtime and Bailey connected on a 21-yard boot, hoisting the Cowboys (8-6) into a three-way tie atop the NFC East with the New York Giants and Washington.
Tony Romo completed 30-of-42 passes for 341 yards and two touchdowns for Dallas, which won for the fifth time in six games.
Roethlisberger, meanwhile, finished with 339 yards and two touchdowns on 24-of-40 efficiency, but couldn't prevent a fourth loss in five contests for Pittsburgh (7-7).
The Steelers fell one game behind AFC North leader Cincinnati as well as one game out of a wild card berth.
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