The Cowboys' Anthony Spencer picked off a tipped Jason Campbell pass just after the 2-minute warning to seal the victory.
Romo passed for 158 yards, completing 15-of-27 attempts, and was picked off twice.
Marion Barber carried 20 times for 99 yards for the Cowboys (7-3).
Shaun Suisham kicked field goals from 45 and 31 yards for Washington (3-7).
Campbell was 24-of-37 for 256 yards.
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