There were chances at both ends before Morton struck first when Ross Forbes, fed by Jai Quitongo, sent a blistering left-foot shot high into the top left-hand corner of Alan Martin’s net.
And the Doonhamers’ keeper was beaten again soon after when Quitongo’s long-range effort soared into the opposite corner.
Queens, though, immediately threw themselves a lifeline when Derek Lyle linked with Stephen Dobbie before beating Derek Gaston with an angled effort.
Gary Harkins sent a header against the post and had another effort saved as the Greenock men tried to restore their two-goal cushion before the break.
But Queens were still firmly in the hunt and they might have equalised on any one of three occasions after the restart but each time Dobbie was thwarted by saves from Gaston.
At the other end Harkins and Thomas O’Ware went close with headers as both sides continued to attack.
Queens threw everything forward in the closing stages but were unable to find that second, equalising goal.