Wesley Matthews added 21 points for the Blazers, who snapped a two-game losing streak.
Manu Ginobili had 14 points for the Spurs, who have lost only eight games this season.
Gary Neal's 3-pointer for San Antonio tied it at 78, but Aldridge's layup and Wesley Matthew's scoop gave Portland its biggest lead of the game so far at 82-78 with 7:03 left.
Aldridge added a 21-foot jumper and Andre Miller had a layup to make it 86-78 before Tim Duncan broke the 8-0 Portland run with a turnaround jumper.
Aldridge's bank shot with 3:33 left made it 90-80 and the Rose Garden crowd was on its feet. Portland led the rest of the way and Aldridge was treated to a standing ovation when he left the game with just over a minute to go.
It was the first game of the Spurs' "Rodeo Road Trip," the team's annual junket while the AT&T Center hosts the San Antonio Stock Show and Rodeo. The next stop on the nine-game trip is the highly anticipated matchup with the defending NBA champion Los Angeles Lakers on Thursday.
The Blazers, who were coming off a lengthy break after losing 88-78 to the Boston Celtics at home last Thursday, were 16-7 at the Rose Garden going into the game.
Portland started Nicolas Batum, who hurt his left knee in the first half of the loss to Boston. An MRI on the knee was clear, bringing a sigh of relief from the Blazers, who have been besieged with knee injuries this season.
Center Greg Oden, the No. 1 draft pick in 2007, who had microfracture surgery on his left knee in November and will miss the rest of the season. All-Star guard Brandon Roy had arthroscopic surgery on both knees on Jan. 17 and is out indefinitely, and center Marcus Camby had surgery to repair a torn meniscus in his left knee on Jan. 20.
Roy and Camby were both at Tuesday night's game. A Rose Garden fan held a sign proclaiming "Depleted Not Defeated."
San Antonio jumped to an early 11-4 lead after Portland was unable to score for the game's opening 4 minutes. The Blazers tied it at 18 before the end of the quarter on Aldridge's dunk, then took a 22-20 lead on his jumper.
The Blazers kept up the pressure but San Antonio kept the edge with 13 points off rebounds compared to Portland's two. Rookie center DeJuan Blair had 14 points and 11 rebounds in the first half as the Spurs led 52-47 at the break.
The teams weren't able to build a lead of more than a handful of points in a tight third quarter. Portland went up 71-88 late but the Spurs answered with Ginobili's 3-pointer that tied it for the start of the fourth.
Earlier Tuesday, Matthews was named to the sophomore squad for the All-Star weekend's rookie-sophomore game. San Antonio's Neal and Blair were both named to the rookie squad.
Notes: A helicopter that was readying to pick up Blazers' owner Paul Allen and take him to his yacht went down in waters off Argentina this week. The two pilots were rescued from the water by Allen's crew and treated for minor injuries. ... Rookie center Chris Johnson was signed by Portland as a free agent on Jan. 24