It was the Suns' third straight victory while the Hawks had a four-game winning streak snapped.
Phoenix came up with 58 points from its bench, which was called on for extra help when Dragic exited the contest.
The Suns took a 10-point lead when Jermaine O'Neal hit a jumper with 2:21 to play and although Phoenix did not score another point it created two turnovers the rest of the way and limited Atlanta to two shots from the field.
Atlanta had been seeking its sixth straight road win. Al Horford led the Hawks with 20 points.