Brandon Carr sealed the win when he picked off a Nick Foles pass and returned it 47 yards for a touchdown.
Romo ended up completing 19-of-26 passes for 209 yards and two TDs, including one to Felix Jones, who compiled 93 all-around yards.
Foles went 22-of-32 through the air for 219 yards and a score in his NFL debut after replacing Michael Vick, who was knocked out of the game with a concussion.
The rookie, however, also lost a fumble that was recovered in the end zone by the Cowboys for a touchdown as the Eagles (3-6) lost a fifth straight game.
Attkisson leaves CBS News, reportedly over network's 'liberal bias'
Beautician charged with giving client fatal silicone butt injection