Stephen Dobbie scored the first and was controversially denied a second after Gary Oliver had levelled the score.
Queens were first to threaten when Ross Fergusson fed Dobbie, whose shot was deflected behind. However, the visitors went closer still when Gary Harkins sent a 20-yard free-kick crashing off the crossbar.
The hosts were next to threaten when Dobbie’s cross was headed into the side netting by Darren Brownlie. Fergusson and Lyndon Dykes also passed up chances as the Dumfries side began to take control.
The opener came when Dykes provided the pass for Dobbie to produce a typically deadly finish.
Morton hit back on the restart when Harkins fed Oliver and the striker drilled a powerful shot low into the corner of the net.
Queens thought they had retaken the lead when Dobbie tried an audacious long-range shot which the home players were convinced had crossed the line before keeper Derek Gaston claimed it.
However, the referee waved play on.