Miller's decisive basket came 13 seconds after Golden State's Stephen Curry had buried a 3-pointer from the left corner to tie the score.
Miller finished with 28 points while Ty Lawson (12 points) and Wilson Chandler (11, 13 rebounds) also made big contributions for the third-seeded Nuggets, who reeled off their 24th straight home victory in taking a 1-0 lead in their first-round Western Conference series.
The sixth-seeded Warriors got 22 points from Klay Thompson while Curry finished with 19 and nine assists.
Golden State's David Lee notched a double-double of 10 points and 14 boards before leaving in the fourth quarter with a likely strained right hip flexor.
2014: The Year in Music [PHOTOS]