Aldridge, Nicolas Batum and Damian Lillard combined to score 91 points for the Trail Blazers, who won their second in a row after having managed just two victories in their first seven games.
Houston led by 10 late in the third period, but Portland ran off 12 points in a row to get in front and had a three-point cushion with time running out in regulation.
Marcus Morris, however, hit a 3-pointer for the Rockets with 16 seconds to go to force the extra period.
Aldridge made 1-of-2 free throws with 1:21 left in overtime to put Portland ahead for good and his jumper with 46 seconds remaining made it a three-point game.
James Harden scored the last of his 29 points at the free throw line with 2 seconds to go and he intentionally missed a second attempt in hopes his team could get the rebound. Houston could not gain possession and dropped the opener of a three-game western trip.
Batum scored 35 points, hitting 13-of-19 shots from the field. Aldridge had 29 and Lillard added 27.
The teams combined to attempt 52 shots from 3-point range -- Portland making 10 and Houston nine.