Darren Collison contributed 20 points in the win, Los Angeles's eighth-straight victory.
"When shots are falling you want to keep shooting," said Griffin.
"It was one of those moments that it felt like everything was going in. When shots are falling from the outside, that opens up everything else."
The Suns have lost two straight and fell to ninth place in the Western Conference playoff race behind Memphis.
The Clippers held a 62-45 lead at the half, and Paul's jumper with just seven minutes left stretched the lead to a game-high 25.
Phoenix followed with a 20-6 run to cut the deficit to 85-74 with less than a minute remaining.
Matt Barnes, who finished with 13 points, nailed a 3 to close the scoring and stake the hosts to a 90-76 lead heading into the fourth.
"You are not going to come back from 25 points and beat these guys," said Suns coach Jeff Hornacek.
"We came out soft. We need to play the way we played in the third and fourth quarters all game. We hung in there. The guys battled. We just can't get in that hole."
In the midst of the run, Tucker and Griffin got tangled up going for a rebound. Tucker appeared to elbow Griffin while on the ground and following a video review, Tucker was ejected for the action.
Marcus Morris' jumper with 6:54 remaining sparked a 19-4 burst that pulled the Suns within six, 105-99, after a Dragic layup.
Dragic would convert another layup with 32.5 seconds left in the fourth to trim the deficit to 109-105, but Barnes responded with two free throws to secure the win.
"We had a big lead and you could see it slipping away," said Clippers coach Doc Rivers. "There was no rhythm. Give them credit; they're fighting for their lives. This was a good win because you are playing a team that needs every win."