The Suns trailed by 13 points heading into the fourth period, but rallied to win their second straight and close within a game of .500.
Los Angeles did not score during the final 2 minutes and fell 2 1.2 games behind the Los Angeles Lakers for the Pacific Division lead.
Phoenix was holding onto a two-point lead when, with the 24-second clock winding down, Brown attempted a 3-pointer. That attempt misfired, but Channing Frye grabbed the offensive rebound and got the ball back to Brown.
Blake Griffin hit 12-of-18 shots from the field and produced 25 points for the Clippers.