Dragic connected for 21 points with seven assists while Brown came off the bench to fire in 19 as the Suns claimed a win for the third time in four games.
Michael Beasley helped out with 16 points and Markieff Morris (13 points, six rebounds, five assists) perpetrated a key steal with 2:16 left in the game for Phoenix.
The Nuggets, meanwhile, saw their four-game winning streak snapped despite 17 points and nine rebounds from Andre Iguodala.
JaVale McGee supplied 16 points in Denver's loss.
Notable deaths of 2014 [PHOTOS]
LeBron James reveals his new Nike shoes