Morrow had six goals in 22 games against Phoenix, then tallied twice in a 1:41 span midway through the first period before capping the first three-goal effort of his career 68 seconds into the second.
"I thought I had a chance for six or seven tonight," Morrow said. "I was getting some bounces and the rebounds were coming to me. I made a couple of good plays off the wall and my linemates found me. That doesn't happen very often, but things seemed to come pretty easy tonight."
The hat trick lifted Morrow's career goal total vs. the Coyotes to nine, two more than he has against St. Louis.
Oliver netted his fourth goal in five games 8:18 into the second to give the Stars a 4-2 lead and Bill Guerin scored the eventual game-winner early in the third.
Marty Turco made 21 saves helping Dallas extend its unbeaten streak to six games.