Houston kept pace with Denver in the battle for the final playoff position in the Western Conference, with both teams moving a game in front of Utah.
Dragic had three 3-pointers and Houston made 7-of-14 as a team from beyond the arc. Memphis made only 1-of-15 from long range.
The score was tied with 3 minutes left in the third period but Houston grabbed a five-point advantage at the end of the quarter and held on from there.
Rudy Gay scored 20 points for Memphis, which has lost in each of its last 12 trips to Houston. The teams split their four games this season.