Smith, Denver's streak-shooting sixth man, scored 12 of his 19 points in the final period and the Nuggets pulled away for a 112-93 victory Saturday night.
"He's an explosive player," Nuggets coach George Karl said. "I thought we ran a couple of plays that got him going."
The Bulls had a 14-point lead in the first half.
"Chauncey went off on us in the second quarter and got them back in the game and then J.R. started hitting shots in the fourth," said Derrick Rose, who led the Bulls with 28 points. "He's tough to guard. It's hard to guard him when he's shooting from everywhere."
The Nuggets came back to tie it at 61 when Billups hit a 3-pointer with 6:09 left in the third quarter. They moved in front for first time since the game's opening moments during their ensuing possession, when Kenyon Martin hit a jump hook over the 7-foot Joakim Noah.
Chicago regained the lead 71-70 on Rose's running jumper, but Anthony made a jumper and Smith connected on a 3-pointer to give the Nuggets a four-point lead heading into the fourth quarter.
Denver took control from there, going on a 14-4 run to start the final period that featured a fast-break layup by rookie Ty Lawson off a steal and a pair of 3-pointers by Smith.
"It started off with defense," Smith said. "We started playing great defense, started running, getting out in transition, getting everybody involved. It became fun and easier to play."
That's also when Smith began to find the range.
"Whenever I get in one of those zones, it usually starts early," Smith said. "This one came late, but I'm not complaining. I'm glad I started making shots. Sometimes, you've just got to roll with the punches, keep shooting and hope you shoot your way out of it."
It was the 15th straight regular-season home win for the Nuggets, including all five this season as Denver bounced back from Friday night's loss in Los Angeles to the Clippers.
"This was a tough win for us, especially coming off a tough loss, tough loss," Billups said. "We feel like we need to get as many games at home as we can, and the only way we can do that is try to win as many games at home as we can. There's going to be some tough nights, but we do focus on establishing a great home court."
Anthony, the NBA's leading scorer, reached 30 points for the ninth time in his first 13 games in helping the Nuggets sweep their two-game season series with Chicago. They won the first meeting earlier this month when officials waved off a potential game-winning basket at the buzzer by Brad Miller, ruling it was an instant too late.
Luol Deng added 22 points, and Noah had 10 for the Bulls, who lost for the 10th time in their last 11 games in Denver.
"I thought it started in the second quarter when Chauncey took over and got them back in," Bulls coach Vinny Del Negro said. "In the third quarter, Carmelo and the fourth quarter, J.R. got going a little bit. They've got three big scorers. I thought our energy, our effort, the physical part of the game; we were not near at the level we need to be at."
Trailing 27-14 after a sluggish first quarter, the Nuggets cut in to the Bulls' advantage in the last 4:34 of the second quarter, when they outscored Chicago 17-7 to pull to 52-49 at halftime.
Billups was at the center of the burst, feeding Nene inside for a dunk and going to the basket himself for 13 of his 14 second-quarter points.
Bulls center Aaron Gray, out all season with a left fibula stress fracture, could be nearing a return. He joined the Bulls in Denver, but Del Negro said he'd like to see him practice a few times and get a better feel for his conditioning before putting him back on the floor. ... Chris Andersen, inactive Friday night because of right knee tendinitis, was back in action for the Nuggets. ... The Nuggets last lost a regular-season game at the Pepsi Center on March 9 to Houston, 97-95.