Anthony scored 19 of his 41 points in the fourth quarter and the Denver Nuggets were helped by Portland's trouble at the free-throw line for a 97-94 victory Thursday night.
Oden missed a pair of free throws with 4.6 seconds left, squandering a chance for Portland to go ahead.
"We came out here and stole the win," Anthony said. "It wasn't just an important win. It was a battle."
Anthony hit a jumper with 4:26 left to tie it at 85, then added another to give the Nuggets their first lead since the second quarter.
It stayed close with the teams trading free throws. But Portland started to have trouble at the line and Nene hooked a shot and Denver led 95-91.
After Oden's misses, Anthony made a pair and Roy missed a 3-pointer at the buzzer for the final margin.
"This is a tough one," Oden said. "I definitely stepped up there with confidence thinking I was going to hit both and it didn't go that way. I put this loss on me. I need to step up and make those."
Roy, who led the Blazers with 30 points, said he would speak to Oden on Friday.
"The biggest thing is to let go," Roy said. "We all miss shots, you've got to forget about it."
The game was slowed considerably by fouls. Combined, the teams attempted 90 free throws. Each team had 20 personal fouls, and the game lasted 2:50.
Anthony made five of seven shots from the floor and nine of his 10 free throws in the fourth quarter.
"My shots were going in. I was trying to attack," Anthony said. "I got to the rim and the foul line."
Both teams had 54 wins last season, but the Nuggets edged the Blazers out of the Northwest Division title.
Denver went to the Western Conference finals, where the team fell 4-2 to the eventual NBA champion Lakers, while Portland fell in six games to the Houston Rockets in the first round of the playoffs.
The Nuggets were playing the second of a back-to-back. Denver opened at home Wednesday night with a 114-105 victory over the Jazz.
The Blazers won their opener 96-87 over the Houston Rockets on Tuesday night.
"I think it [the fouls] benefited them a bit more coming off a back-to-back," Roy said. "They were tired but anytime you slow the game down that much, it gives them a chance to recover."
Denver jumped out to an early lead, holding a 24-17 advantage after the first quarter. The Blazers battled back and led 43-42 at the half.
Anthony amassed 20 points in the half. His layup midway through the third quarter narrowed the Nuggets to within 53-52. Roy's jumper kept Denver at bay.
Billups hit a 3-pointer to make it 61-59, before Chris Andersen made a pair of free throws for Denver to tie it at 61 with 1:23 left in the third.
The Nuggets kept it close until Roy hit a running 28-footer at the buzzer to put the Blazers ahead 67-62.
Portland couldn't stretch it further until Rudy Fernandez hit a 3-pointer over a charging Anthony to make it 81-74. But the Nuggets silenced the Rose Garden crowd with consecutive layups from Anthony and Lawson.
"Anthony was simply amazing," Nuggets coach George Karl said. "We rode Carmelo."
Anthony had 30 points in Denver's opener.
Denver's J.R. Smith served the second game of his season-opening seven-game suspension for pleading guilty to reckless driving over the summer. ... Former University of Oregon quarterback Joey Harrington attended the game. ... Team owner Paul Allen, who reportedly underwent heart surgery over the summer, also attended the game.