Turner had 20 carries for 102 yards and Matt Ryan had a season-high 342 yards passing for the Falcons (8-0), who took their first lead on Turner’s TD with 14:21 remaining.
Tony Romo completed 6 of 6 passes for 78 yards, including a 21-yard scoring pass to Kevin Ogletree, on the Cowboys’ touchdown drive later in the fourth.
The Falcons then worked the clock, holding the ball for 5 minutes, 4 seconds, leaving only 17 seconds left after Bryant’s 32-yard field goal.
Romo completed three short passes before the game ended on a pass to Felix Jones, who was tackled at the Atlanta 22.
Roddy White became the leading receiver in Atlanta history late in the third quarter with his 574th career catch.
The Cowboys had the early upper hand.
The Falcons seemed to have few answers for outside linebacker DeMarcus Ware and nose tackle Jay Ratiff, stalwarts of Dallas’ defense who were pushing back Atlanta’s offensive line and collapsing the pocket around Ryan.
Midway through the second quarter, Ware sacked Ryan for a 7-yard loss and caused a fumble, but White kept the possession alive by falling on the ball at the Dallas 48.
Dallas was playing without running back DeMarco Murray, who missed his third straight game with a sprained right foot. Atlanta linebacker Sean Weatherspoon was sidelined by a sprained right ankle, and defensive tackle Peria Jerry left with a knee injury in the second quarter.
Atlanta, which was coming off a 30-17 victory last week at Philadelphia that was one of its most complete games under fifth-year coach Mike Smith, cut the lead to 6-3 on Bryant’s 45-yard field early in the second quarter.
Bailey missed a 54-yarder wide left on Dallas’ ensuing possession.