Dallas has won a record 49 straight games when leading after two periods but saw that streak jeopardized when Ziggy Palffy scored with 2:49 left in regulation.
As the clock wound down in overtime, Morrow carried the puck to the left faceoff dot and fired a wrist shot past the glove side of goaltender Felix Potivn.
Goalie Marty Turco finished with 24 saves and extended his personal winning streak to five games. Jere Lehtinen and defenseman Richard Matvichuk had the other goals for Dallas, which improved to 3-4-7 in overtime.
Adam Deadmarsh had the other goal and Potvin made 28 saves for the Kings, who are 0-2-1 in their last three games.