The Blazers have won seven of their last eight games and they've dropped just two games at home this season.
But the Clippers, playing the second of a back-to-back, mounted a challenge after trailing by as many as 11 points early in the second half. Blake Griffin had 35 points and 11 rebounds, while Chris Paul finished with 34 points and 16 assists.
Aldridge started for the Blazers even though he was a game-time decision because he had his wisdom teeth pulled. He hit a fadeaway with 39 seconds left in overtime and Portland went ahead 110-109.
Griffin missed a layup on the other end before Nicolas Batum made a pair of free throws to make it 112-109 with 26.5 seconds to go. After Griffin dunked to narrow it again, Damian Lillard made two free throws for Portland.
Portland clung to a 114-112 lead with 13.4 seconds left and padded it with Wesley Matthews' foul shots. Paul missed a 3-point attempt as time ran out.
The Blazers were coming off 110-107 victory New Orleans last Saturday.
Earlier Thursday, the NBA said its referees made a mistake when they ejected Griffin in the Clippers' 105-103 loss to the Warriors on Wednesday night.
Griffin got his second technical with 10:43 left after scuffling with Andrew Bogut, following Warriors forward Draymond Green to the showers after they got into it at the end of the third.
"A common foul should have been called on Griffin for initially attempting to dislodge the Warriors' Andrew Bogut and a technical foul should have been assessed to Bogut for grabbing Griffin by the shirt and wrestling with him," Rod Thorn, the NBA's president of basketball operations.
Clippers coach Doc Rivers said before the game against the Blazers that he thought Griffin handled the situation well.
"Everyone is a target at some point in their career," Rivers said. "You just have to play through it."
It was the second ejection for the Clippers in as many games. Forward Matt Barnes was ejected with 56 seconds left in the third quarter of a 120-116 OT win over the Minnesota Timberwolves on Sunday night. Barnes fouled Kevin Love.
"It's a long season. We're in a bad ejection rut right now," Rivers said. "We'll get out of it."
Mo Williams made back-to-back 3s to give Portland a 34-22 in the first half. The Clippers responded with a 9-0 run to close to 34-31, but Aldridge scored six straight points in an 8-2 run and Portland stretched the margin to 42-33.
Paul's 3-pointer that closed the Clippers to within 47-46 at the intermission.
Portland pulled ahead 61-43 in the third quarter but Aldridge picked up his fourth foul and the Clippers went on an 8-0 run to make it 61-59.
Dorell Wright hit a 3-pointer that gave the Blazers an 84-75 lead early in the fourth quarter.
But the Clippers clawed back in, and Paul's pull-up jumper made it 97-96 for Los Angeles' first lead of the second half. After Aldridge's tip-in, Paul added another jumper to give the Clippers a 99-98 lead then nailed a step-back 21-footer from the corner with 9.3 seconds left.
Undaunted, Batum hit a 3-pointer to tie it at 101 with 5.3 seconds to go. Paul missed a 17-footer at the jumper to send the game into overtime.
The game was the first of three meetings between the two teams this season. The Clippers claimed last season's series 3-1. ... The Clippers' 14 first-quarter points are the fewest they've scored in the first quarter. ... DeAndre Jordan finished with 19 rebounds before fouling out in overtime. ... Barnes also fouled out in OT.