The Falcons, who are enjoying their best start to the season, were helped by a dogged defensive display, while quarterback Matt Ryan produced a superb performance.
Ryan threw for a season-high 342 yards, although he was unable to connect with any scoring passes.
Instead Michael Turner, who rushed for 102 yards, scored a touchdown off a three-yard run, while Matt Bryant landed four field goals.
The result means Atlanta are now 8-0 for the campaign and are the only unbeaten franchise left.
The Cowboys, on the other hand, drop to 3-5 despite pushing the Falcons hard throughout.
Dan Bailey landed two early field goals to keep Dallas level at 6-6 at half-time before Atlanta were able to move clear.
Quarterback Tony Romo connected with receiver Kevin Ogletree for a 21-yard touchdown with just over five minutes left but Dallas were left with just too much to do.
Romo completed 25 of 35 for 321 yards, while Ogletree racked up 96 running yards.