Cleveland lost despite LeBron James having a triple-double by the end of regulation, finishing with 43 points, 13 rebounds and 15 assists.
Anthony matched him shot for shot, hitting the winner on a deep jumper over an outstretched James with 1.9 seconds left to give Denver a win that opened a 1-1/2 game lead atop the Northwest Division.
Kenyon Martin had 18 points and a season-high 17 rebounds for the Nuggets, who had plenty of motivation to begin the second half of its season with a win.
Nuggets coach George Karl was on the bench despite battling nausea and headaches in recent days as a result of chemotherapy. Karl coached his first game since disclosing Tuesday he has neck and throat cancer.
The Cavaliers had a chance to win it at the buzzer, but James slipped while receiving the inbound pass and his 3-pointer rattled off the rim. An exhausted James, who played 47 minutes, missed his final four shots in overtime.
His triple-double was the third of the season and 27th of his career. His 15 assists tied a career high.
Cleveland made just 23 of 40 free throws.
The Cavaliers were undermanned since they traded Zydrunas Ilgauskas on Wednesday and didn't yet have Antawn Jamison available - the player they received in return from Washington in the three-team deal. The Cavs introduced Jamison on Thursday, but he wasn't activated in time for the game. The team is hopeful he can be in the lineup Friday at Charlotte.
In Los Angeles, Ray Allen scored 12 of his 24 points in the third quarter and made 10 of his first 11 shots as Boston edged Los Angeles.
It was the defending NBA champions' first loss in five games since Kobe Bryant sustained a sprained left ankle.
Kendrick Perkins had 13 points and 14 assists for the Celtics, who were busy before gametime. They acquired guard Nate Robinson and Marcus Landry from the New York Knicks in exchange for Eddie House, J.R. Giddens, Bill Walker and a future conditional second-round draft pick.
Paul Gasol scored 22 points to lead Los Angeles.
The Lakers had won the previous three meetings with the Celtics after losing the 2008 finals to them in six games.
On Thursday they led 84-80 in the fourth quarter but went 4 minutes, 48 seconds without a basket to cede the lead.
The Laker had one final opportunity on the buzzer, but Derek Fisher missed a 21-footer.
