Mattias Norstrom and Jeff Halpern had power-play goals in a 20-second span of the second period to draw Dallas even at 4. The Blues had taken a 4-2 lead earlier in the period on goals by Martin Rucinsky, Lee Stempniak and Matt Walker.
Morrow put the Stars in front for good when he fired a loose puck from close range past goaltender Hannu Toivonen.
Brad Winchester's goal with 4:17 left stretched Dallas' lead to 6-4, but Erik Johnson struck on the power play with 2:16 remaining to pull St. Louis within one.
Patrik Berglund added a goal and Paul Kariya had four assists for St. Louis, which has lost both of its preseason games.