Gary Harris poured in a game-high 32 points and Will Barton added 23 as the Nuggets, coming off a 108-102 win at Memphis the night before, won for the sixth time in their last eight games.
LeBron James recorded a 25-point, 10-rebound, 15-assist triple-double for the Cavaliers, who have lost four of their last six games.
The clubs meet again in Denver on Wednesday.
After trailing by as many as 15 points in the third quarter, the Cavaliers rallied to within 115-114 by holding the Nuggets scoreless for eight straight possessions, with the Nuggets missing seven shots in that span.
But after two free throws by James with 2:40 left got Cleveland within one, Millsap connected on his critical hoop, sending the Nuggets on an 11-0 burst that produced the victory.
Harris used 6-for-10 shooting on 3-pointers to account for a majority of his 32 points. The Nuggets went 19-for-35 from beyond the arc, outscoring the Cavaliers 57-27 on 3-pointers.
Barton collected a team-high eight rebounds and seven assists to complement his 23 points, while Nikola Jokic had a similarly well-rounded game with nine points, seven rebounds and a team-high eight assists.
Wilson Chandler and Jamal Murray added 16 points apiece and Millsap finished with 11 for the Nuggets, who will go for a 3-0 trip Tuesday at Dallas.
JR Smith backed James with 19 points for the Cavaliers, who fell to 5-4 since reshaping their roster at the trade deadline.
James lost for the 15th time in 27 career games against Denver. He has a losing record against just three teams, the others being Houston and San Antonio.
George Hill (17 points), Larry Nance Jr. (14), Cedi Osman (13), Tristan Thompson (12) and Jordan Clarkson (11) also scored in double figures for the Cavaliers, who never led in the final three quarters.
Playing the second night of a back-to-back, the Nuggets stormed out to a 10-point lead in the first quarter and never looked back.
They went up by as many as 14 in the second period and 15 in the third, at 99-84, before Hill (two) and Clarkson (one) combined for three 3-pointers in a 13-4 period-closing run that got Cleveland back within 103-97.
--Field Level Media
Denver | Cleveland | |
Gary Harris 32 | Scoring | LeBron James 25 |
Nikola Jokic 8 | Assists | LeBron James 15 |
Will Barton 8 | Rebounds | LeBron James 10 |
Gary Harris 6 | Free Throws Made | LeBron James 8 |
Gary Harris 1 | Steals | LeBron James 2 |
Wilson Chandler 1 | Blocks | Rodney Hood 1 |
Team | Points | FG% | 3PM-3PA | FTM-FTA | Assists | Rebounds | Blocks | Steals | Turnovers |
Denver | 126 | 54.7 | 19-35 | 13-15 | 35 | 35 | 3 | 4 | 9 |
Cleveland | 117 | 52.3 | 9-24 | 16-21 | 25 | 41 | 2 | 5 | 9 |