The teams were tied at 67 when Spanoulis made a 20-footer to give the Rockets the lead for good. Steve Novak added a 3-pointer with 37.7 seconds left.
Ndudi Ebi made two free throws with 32.3 seconds remaining to get Dallas within three. The Mavericks had a chance to tie it with 10.8 seconds left, but Darius Washington shot an airball on a 3-point attempt from the right wing, and time ran out before Dallas could get off another attempt.
Houston center Yao Ming had eight points, eight rebounds and two blocked shots in his first game of the preseason.
Luther Head scored 13 points and Novak finished with 12 for the Rockets (2-1), who rallied from a 14-point third-quarter deficit.
Josh Howard had 22 points and Maurice Ager added 12 for the Mavericks (0-4).
Dallas was without six of its top eight players, resting stars Dirk Nowitzki, Jason Terry and Jerry Stackhouse.
The Mavericks got a pair of 3-pointers from Howard during a 13-4 spurt to begin the second half, pushing their lead to 50-36.
But the Rockets went on a 16-4 run over the final 6:22 of the third quarter, fueled by Head's five points, to narrow the deficit to 54-52 entering the final quarter.