Rose made 11 of his 22 shots, including two big layups in the final minute that helped the Bulls to their third consecutive win. Joakim Noah provided 13 points and 14 rebounds.
"We're grateful for this win, it just goes onto our record," Rose said. "But we know when we're in this position like this, we have to find ways to just put teams down, step on them and make sure they don't have any confidence to come back."
Antawn Jamison scored 21 points to pace the Cavaliers, whose losing streak reached six games after they missed two potential game-tying shots in the closing moments.
Anderson Varejao added 17 points and 12 rebounds in the setback.
"We didn't get the outcome that we were looking for, but that's something to build on, because I thought our guys played hard for the first time in a while," said Cavaliers head coach Byron Scott.
Cleveland led for most of the fourth quarter, and held an 80-74 edge after Jamison made a free throw with 3:01 left to play. However, the Bulls evened the game with two Ronnie Brewer baskets and two Rose free throws.
With just more than a minute to go, Cavaliers guard Daniel Gibson missed a three, and the Bulls went ahead on the ensuing possession. Rose drove down the middle of the lane, bounced off Mo Williams and sank a floater with 46.9 seconds left.
A pick-and-roll left Varejao wide open for a layup at the other end, knotting the score at 82, and the Cavaliers had an opportunity after Rose missed a jumper from the foul line. Varejao was fouled on the play and made 1-of-2 free throws with 26.2 ticks left.
Rose again went to work against Williams, driving past him down the left side. Rose then cut to the middle of the paint and threw in a layup with his right hand as he was fouled. He made the free throw for an 85-83 edge with 19.6 seconds remaining.
Cleveland still had plenty of time to pull even or take the lead, but after Williams missed a baseline jumper, Gibson's bank shot bounced off the rim, and the Bulls secured the win at the foul line.
Gibson finished with 14 points for the Cavaliers, while Williams ended with 13 points and 10 assists, but made only 5-of-18 shots.
Chicago opened the game on a 10-0 run before taking a 21-15 lead to the second quarter. Cleveland used an eight-point run late in the half to briefly move in front, but the Bulls countered with an eight-point burst of their own to take a 41-35 lead into the break.
The third quarter was close until Cleveland ended it with nine consecutive points. Williams' runner with time expiring gave the Cavaliers a 68-60 lead moving to the fourth.
But sharpshooter Kyle Korver kept Chicago in the game, opening the final quarter with a three-pointer, a technical free throw and a jumper. He had not scored to that point, but netted the Bulls' first 12 points of the quarter.
Luol Deng finished with 13 points for the Bulls, who shot 45 percent...Cleveland made 38 percent of its shots...The Cavaliers begin a four- game road trip Saturday in Houston. They will also make stops in Oklahoma City, Miami and Indiana.