Anthony scored 41 points in a foul-filled game between teams that had won their season openers.
Denver was clinging to a one-point lead when Chauncey Billups missed a jumper with 13 seconds to play. Portland worked the ball to Greg Oden for the potential winning basket but he was fouled by Kenyon Martin.
Oden then missed both his foul shots with 5 seconds left and Anthony grabbed the rebound. Anthony was fouled by Martell Webster and made both his attempts to provide the final margin.
Brandon Roy missed an attempt to tie it at the buzzer.
Billups added 22 points for the Nuggets while Portland was led by Roy with 30 and Rudy Fernandez with 22.
The teams combined to make 71-of-90 free throws.
Notable deaths of 2014 [PHOTOS]