Minnesota's Kevin Martin missed a layup at the buzzer, completing a collapse that saw the Suns end the game by outscoring the Timberwolves 7-0.
Corey Brewer made 1-of-2 free throws for Minnesota with 1:51 to go, giving the Timberwolves a 103-97 lead.
Minnesota could not score another point and gave Phoenix a chance to take the lead when Ricky Rubio suffered a turnover with 15 seconds to go.
That led to Green's game-winning shot, giving him 14 points for the contest.
Martin delivered 20 points and Nikola Pekovic tacked on 17 and 12 rebounds for Minnesota.