Morrow, who took over the captaincy from Mike Modano on Friday, won the game off an assist from Modano after the former captain tied it with 6:12 left in regulation.
The Stars (3-4) overcame a 3-0 deficit to force overtime. Steve Ott pulled Dallas within two goals midway through the second period, and Stu Barnes made it 3-2 with 7:58 left in the third.
Vaclav Prospal, Dmitry Afanasenkov and Tim Taylor scored for Tampa Bay (2-1-3), and Marc Denis and Johan Holmqvist combined to make 33 saves.
Marty Turco made 17 saves for the Stars.