Portland led by five early in the fourth quarter before Marcus Morris hit consecutive 3-pointers to give Houston a 69-68 lead.
"We didn't get quality shots," Houston coach Kevin McHale said. "We got to get better at that."
Dozens of fake beards made to resemble Harden's trademark look dotted the sellout crowd of 18,140. The Rockets were selling them at the team store on Saturday night to take advantage of the excitement created by their newest acquisition. Houston's mascot Clutch even got in on the fun during a break in the third quarter and donned a fake beard, too.
Harden said he tweaked an ankle in the game and it slowed him a bit.
"But I don't like to make excuses," he said. "I had opportunities to make plays and to make shots and I just didn't."
Portland used a 12-4 run at the beginning of the second half to tie it at 49 as the Rockets struggled to find a rhythm in the third quarter.
Harden didn't score after halftime until he converted a three-point play with just over four minutes remaining in the third, giving Houston a 58-57 lead.
The Trail Blazers answered with seven straight points.
An alley-oop dunk by Morris got Houston rolling again and it trailed 64-63 entering the fourth quarter.
Harden grabbed a microphone and welcomed fans to the game before the national anthem, and got the team going by scoring Houston's first six points. He had 10 at the end of the first quarter to help the Rockets lead 19-17.
Houston opened the second quarter with a 10-0 run. The Trail Blazers' first points in the second came on a 3 by Matthews less than eight minutes before halftime.
That was the first of seven straight points for Portland before Lin got Houston going again by hitting a nifty one-handed layup and the foul shot.
The Rockets led 45-37 at halftime.
NOTES: Omer Asik, who had a career-high 19 rebounds on Friday, had 15 rebounds for Houston and Morris finished with 13 points. ... Chandler Parsons had 12 rebounds for the Rockets. ... Aldridge has scored at least 20 points in each of his last nine regular-season games against Houston.