The Cowboys gave themselves a chance to pull off the upset when Dan Bailey kicked his third field goal of the game with 2:42 to play, a 26-yarder that put Dallas in front by three.
That gave Brady plenty of time, however, to pull out the victory. He completed 8-of-9 passes on the winning drive, the last one going 8 yards to Aaron Hernandez for the score.
Brady completed 27-of-41 passes for 289 yards and two touchdowns, but he also was intercepted twice.
New England did not score in the second half until the Brady-to-Hernandez pass.
Romo also completed 27-of-41 passes for 317 yards and an interception.
The Patriots moved into sole possession of the lead in the AFC East with a 5-1 record, a game in front of Buffalo. The Bills dropped out of a share of the top spot Sunday with a loss to the New York Giants.
Dallas is 2-3. All of the games in which the Cowboys have played this year have been decided by four or fewer points and Dallas held leads in the fourth quarter in all of them.