Chris Paul added 20 points, including a 32-foot 3-pointer just before the shot clock expired to put the Clippers up for good 108-106 with 1:48 to play.
Los Angeles, winning the last four on a five-game trip, clinched the Pacific Division title with the victory.
"It's nice but it doesn't really mean much," Paul said. "We won the division title last year and lost in the first round. Last year, we would have gone into the game knowing we had a chance to win the division. Now we don't care, it's all about the bigger picture."
Eight Suns reached double figures, led by Marcus Morris with 16 points.
Phoenix would have been alone in the No. 7 spot in the West with a victory, only a half-game behind Golden State. Instead, the Suns fell into a three-way tie with Memphis and Dallas for the last two playoff spots in the West. Phoenix would lose the tiebreaker the way things stood Wednesday night.
The defeat followed a blowout loss to the Lakers in Los Angeles on Sunday.
"We had this game almost in our pockets," Goran Dragic said. "But now is not the time to think like that, because we still have to have our confidence up to play those last seven games."
The Suns used a 22-5 run to lead 91-74 with 1:49 left in the third period and were up 91-78 entering the fourth.
Griffin had been listed as doubtful for the game with back spasms.
Ex-Sun Jared Dudley made two free throws with 8.1 seconds to play to clinch the victory.
Dragic and Gerald Green scored 15 points apiece for the Suns, but Dragic was only 2 for 11 from the field, 0 for 5 from 3-point range. P.J. Tucker, whose rebounding was a big factor in the Suns' third-quarter run, had 10 points and 11 rebounds.
With Griffin and Paul on the bench, the Clippers scored 11 in a row early in the fourth quarter -- the last five by Willie Green -- to cut Phoenix's lead to 92-85. Dudley's 3-pointer with 5:57 to go cut it to 97-94.
"You know, we won the game because of our second unit," Los Angeles coach Doc Rivers said. "Honestly, when we were down 17, I was thinking if we didn't cut into it, I was just going to keep them out on the floor, but then they cut into the lead so quickly."
Dudley's two free throws put Los Angeles ahead 105-104 with 2:38 left. Eric Bledsoe's 9-foot floater gave the Suns their last lead, 106-105 with 2:15 left.
Phoenix's defense was strong on the next possession, and just before the clock was to expire, Paul hit from far beyond the arc. Griffin's rebound basket with 49.5 seconds left made it 110-106. Bledsoe scored on the layup to cut it to 110-108.
The Suns had a chance to tie it, but Collison blocked Bledsoe's layup from behind. The ball went out of bounds off the Suns and Dudley's free throws iced the victory.
"We can't afford to have lulls against good teams like this," Phoenix coach Jeff Hornacek said.
The Suns broke away from a 69-69 tie with their 22-5 outburst. Tucker had three rebound baskets in the run. He was fouled on his final one, capping the big run with a three-point play that gave Phoenix its biggest lead, 91-74, with 1:49 left in the period. The Clippers scored the next four to trail 91-78 entering the fourth.
Phoenix led 62-60 at the break.
Paul has eight consecutive games with at least three steals. ... Collison has scored at least 20 points in consecutive games for the first time in four years. ... Bledsoe drew a technical foul for slamming his hand on the scorer's table after being called for an offensive foul, his third foul of the game, with 10:39 left in first half. ... Jamal Crawford, in and out of the Clippers lineup lately, was out again with a sore left Achilles tendon. ... Only two of Phoenix's seven remaining games are at home.