Harden scored Houston's final six points and had 17 of his 30 in the fourth period while boosting the Rockets to their fifth consecutive win.
Houston went in front by a point with 2:14 to play when Harden drove to the basket and produced a layup, but the Timberwolves regained the advantage 20 seconds later on a layup by Andrei Kirilenko.
Harden then hit a running layup with 39 seconds left to put the Rockets ahead again and, after Houston's Jeremy Lin stole the ball from J.J. Barea, Harden hit another layup for a three-point lead.
Minnesota's Alexey Shved missed a 3-point attempt in the final second that would have forced overtime.
Barea led Minnesota with 18 points while Kevin Love hit just 3-of-14 from the field and scored only seven points. Love had missed the previous game with an eye injury.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys