Derrick Rose had 36 points for the Bulls, who lost their second straight. Chicago has not lost three in a row this season.
Aldridge's 20-foot jumper with 1:11 left made it 100-93 and appeared to seal it for the Blazers, but Rose answered with a 3-pointer with just more than a minute to go.
Aldridge hit a turnaround jumper and Andre Miller made a pair of free throws with 30 seconds left to make it 104-96. Luol Deng had a reverse layup on the other end, but the Bulls could not catch up.
Chicago has lost four straight and 11 of their last 13 at the Rose Garden.
The Bulls were coming off a 101-90 loss at Golden State that snapped a six-game winning streak. They dropped to 11-12 on the road.
The injury-depleted Blazers were short again because forward Dante Cunningham fractured his right eye socket on Saturday in Portland's 111-105 victory at Cleveland. Cunningham wore special eye gear and was listed as probable against the Bulls, but he did not play.
Just before the game, Brandon Roy told The Oregonian newspaper that he could return to the lineup as early as this weekend. The three-time All-Star had arthroscopic surgery on both of his knees three weeks ago. The team said Roy was cleared to return to practice late last week.
Roy watched Monday night's game on the bench dressed in street clothes, as did center Marcus Camby, who had arthroscopic surgery on his left knee Jan. 20.
The Bulls went on a 9-0 run to open the second quarter, capped by Ronnie Brewer's layup to make it 35-28. Portland took the lead at 44-43 after Rudy Fernandez hit a pair of free throws and it was back and forth for the rest of the half.
Wesley Matthews scored on a fast-break dunk off an off-balance, one-handed hurl from Fernandez that made it 50-49, but Rose made one of two free throws as time wound down and the game was tied at halftime.
The Blazers came out strong in the second half, going up 63-56 early after Aldridge's dunk. He had 16 points in the third quarter alone.
The Bulls pulled within 75-64 early in the fourth quarter on Deng's jumper, but the Blazers extended it again when Fernandez hit a 3-pointer to make it 80-74.
Brewer's dunk narrowed it to 82-81 with 7:20 left but Chicago was unable to go ahead. Rose had a decisive dunk that pulled the Bulls to 91-87, but again the Blazers kept Chicago at bay.
Aldridge's alley-oop dunk from Miller put Portland up 98-91 with 2:17 left.
Notes: Oregon basketball coach Dana Altman was at the game. Altman coached Chicago's Kyle Korver at Creighton in Nebraska. ... Portland shot 78.6 percent from the floor in the first quarter, but led just 28-26.