Aldridge hit two free throws for the final points of the game and the Nuggets wound up going to Danilo Gallinari for a 3-point attempt to win it.
Gallinari, who came to Denver this week in the trade that sent Carmelo Anthony to New York, missed the long-range attempt and the ball went out of bounds off the Trail Blazers with 1.7 seconds to play.
Denver's Arron Affalo got off a desperation 3-pointer from the corner that failed to go.
Portland made it into overtime thanks to Brandon Roy's 3-point shot with 6 seconds to play in the fourth period.
Aldridge scored 25 points for Portland and both Roy and Rudy Fernandez had 17.
Gallinari made 15-of-17 free-throw attempts and led Denver with 30 points. Wilson Chandler had 20 and Affalo added 19.