Chris Paul added 20 points, including a 32-foot 3-pointer just before the shot clock was to expire to put the Clippers up for good 108-106 with 1:48 to play.
Eight Suns players reached double figures, led by Marcus Morris' 16 points.
Phoenix fell into a three-way tie with Memphis and Dallas for the last two playoff spots in the West. Phoenix used a 22-5 run to lead 91-74 with 1:49 left in the third period and was up 91-78 entering the fourth.
Griffin had been listed as doubtful for the game with back spasms.