Turner's score put the Falcons ahead 13-6 and Matt Bryant added two fourth-quarter field goals as NFC South-leading Atlanta (8-0) continued the best start in its franchise history.
Turner finished with 102 yards on 20 carries, while Matt Ryan connected on 24-of-34 passes for 342 yards with no interceptions.
Julio Jones made five catches for 129 yards and Roddy White added seven grabs for 118 yards for Atlanta.
Tony Romo went 25-of-35 for 321 passing yards with a late TD toss to Kevin Ogletree for the Cowboys (3-5), who lost their second straight and third in four contests.
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