Aldridge fouled out with about a minute left in overtime and Lillard, who was making his playoff debut, took over. He scored the next five points for Portland and put the Trail Blazers on top by one point with a pair of free throws with 17 seconds left.
Game 2 is Wednesday night in Houston.
Aldridge, who was playing in his home state, also had 18 rebounds and two blocks. James Harden and Dwight Howard each scored 27 points for Houston.