Atlanta won for the first time in three games and improved to 1-1 on a five-game road trip. The Hawks became the fifth team in the NBA this season to reach 10 road victories.
Josh Smith led the Hawks with 30 points and 17 rebounds while Green added 20.
Atlanta led by nine with 2:20 to play but Phoenix cut the deficit to two when Steve Nash made a pair of free throws with less than 10 seconds left to play.
Green then hit his two foul shots to increase the Atlanta advantage and the Suns' hopes died when Nash missed a 3-point try.
Nash led Phoenix with 22 points and 16 assists. Grant Hill had 18 points in the Suns' third straight loss.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged