DeMarre Carroll and Paul Millsap put in 19 points apiece for Atlanta, which has dropped three consecutive tilts on the heels of a five-game winning streak.
The game was tied 59-59 midway through the third quarter before Phoenix seized control of the contest with a 10-2 run. Channing Frye had five points during the surge, which he capped with a 3-pointer for a 69-61 advantage late in the third.
The Suns, who outscored Atlanta 35-23 in the third, headed into the final frame with a 77-68 spread.
Mike Scott finished off a quick 6-0 spurt with a layup to pull the Hawks within 81-77 with 8:44 left, but they never got any closer. Green netted seven points during a 9-3 Phoenix swing a few minutes later and his trey staked the Suns to a 92-82 margin with 5:14 remaining.
Earlier, Frye hit a pair of triples, while Dragic and Bledsoe also knocked down one apiece as the Suns raced out to a 15-7 start five minutes in. Atlanta rallied with a 12-4 run to tie the game at 23-23 before heading into the second with a 25-23 edge.
Atlanta led 45-42 at the break.