In a game that was tight from start to finish, Bledsoe gave the Suns their second win in as many tries this season.
Bledsoe scored 14 points in the final 4:02 of the fourth period, but he missed a free throw with 30 seconds remaining that would have given Phoenix a four-point advantage.
Gordon Hayward then gave Utah a chance by making a 3-point shot that tied it up with 26 seconds to play.
After a time out, the Suns went to Bledsoe and he came through with the last of his 18 points. He made only 4-of-15 from the field, but hit 8-of-10 from the foul line. Miles Plumlee had 13 points and 13 rebounds for the Suns, while P.J. Tucker scored 12.
Utah, which has lost both its games this season, was led by 20 points from Enes Kanter.