The Nuggets, who have spent an NBA-high 17 of 24 games on the road this season, improved to 12-12 on their return to Denver, while condemning the Grizzlies to a third consecutive loss.
Andre Iguodala had a big night for the Nuggets with 20 points and seven assists, while benchman Andre Miller contributed 18 points.
"He is amazing with some of the stuff he does and how he gets it done," Nuggets coach George Karl told reporters of the 36-year-old Miller.
"We don't win this game without Andre Miller, without a question.
"It was a slow-down, drag, pull teeth, punch, grab, hold, kind of an old-school basketball game.
"I think it is a really classy win for us. It is a tough-minded win against a team that I think is going to be very, very successful this year."
Memphis (14-6) had a chance to tie the contest in the closing seconds but top scorer Rudy Gay's (21 points) three-point attempt failed to drop as the Grizzlies slipped two games behind San Antonio in the Southwest Division.
"Honestly it shouldn't have gotten to that point," Gay said of the late shot that went agonisingly close.
"I feel like we could have won this game. But whether that shot went down or not I think it was a couple of crucial plays that happened before, earlier in the fourth quarter.
"But now this is where we show who we are. We're going to find a way to get out of this."
The win was Denver's 29th home triumph over Memphis in 33 contests, continuing an impressive dominance in Colorado. (Reporting by Ben Everill in Los Angeles; Editing by John O'Brien)