Stars center Mike Ribeiro drew the Kings' defense in as he skated up the middle and dropped a pass to Morrow, who collected the puck and beat Jonathan Bernier high to the short side for the game-winner.
Jamie Benn's shorthanded, unassisted goal off a turnover by Kings defenseman Drew Doughty 5:20 into the third period produced a 3-3 tie for the Stars, who used the victory to vault past Phoenix and Calgary into the Western Conference's fifth seed.
Ribeiro notched a goal and two assists, Trevor Daley scored a goal and Kari Lehtonen stopped 20-of-23 shots for Dallas.
Justin Williams tallied a goal and an assist for Los Angeles, which retained the eighth and final playoff spot in the West.
Willie Mitchell and Kyle Clifford also scored and Bernier made 18 stops for the Kings.