Charles Dunne headed home inside three minutes of his debut and Chris Cadden’s goal put Motherwell in cruise control before Gary Harkins was sent off in the 39th minute.
Richard Tait added a third and Moult struck moments after coming off the bench as Motherwell made it nine goals in two Group F matches, passing and moving around the Fir Park pitch in some style.
Motherwell could have scored even earlier after goalkeeper Trevor Carson’s huge kick set up Craig Tanner but Derek Gaston denied him.
The opener came from the resulting corner as Carl McHugh headed Tanner’s delivery into the danger zone for defender Dunne to nod inside the near post.
McHugh came close with a diving header from another corner by Tanner, who set up the second goal for Cadden in the 25th minute with a perfect lay-off which allowed the midfielder to slot into the corner.
Morton’s task became even more difficult when Harkins was shown a straight red card for a high challenge on Tanner.
Gaston twice looked lively to deny Ryan Bowman and saved well from Cadden following Tanner’s backheel.
Motherwell were utterly dominant and Tait drilled home from 16 yards midway through the second half after McHugh’s curling effort had sailed over the motionless Gaston and bounced off the bar.
There was no respite for the Championship side as Stephen Robinson immediately brought on strikers Alex Fisher and Moult and the latter soon curled inside Gaston’s near post from the left corner of the penalty box in the 72nd minute.
Gaston made an impressive double stop from Moult and Gael Bigirimana to keep the score down.