Barry Smith's side forgot about their troubles at the bottom of the SPL as their second-half surge saw off Allan Moore's men.
Strikes after the break from Lewis Toshney, Colin Nish, John Baird and Declan Gallagher saw them go through to the last eight after Michael Tidser had cancelled out Jim McAlister's opener to see the sides go in level at half-time.
Dundee carved out the first real goal-scoring opportunity in 13 minutes when McAlister's corner was met by Baird and his header was turned around the post by visiting goalkeeper Derek Gaston.
A few minutes later, they went on the offensive again in search of an opening goal to settle any lingering nerves.
This time winger Nicky Riley tried his luck with a low 25-yard snap-shot which flashed just inches past Gaston's right-hand post.
It was all Dundee and midway through the first-half Baird was presented with another opportunity to break the deadlock.
Strike partner Nish helped the ball on for Baird to run through on goal but from an acute angle he fired the ball into Gaston's side-netting.
Soon after Morton had their first real chance when they were awarded a free-kick 25-yards from goal.
Martin Hardie stepped up and forced Rab Douglas to fling himself full-length to claw away his effort.
At the other end, Nish wasted a chance with the goal at his mercy in 28 minutes.
Riley's cross eluded everyone inside the box before Nish popped up at the far post but somehow he swiped at fresh air .
However, after 29 minutes McAlister saved Nish's blushes as he finally put the home side ahead.
Brian Easton's flighted ball forward was flicked on by Nish perfectly for McAlister to run onto, and the midfielder made no mistake against his former club as he steadied himself before angling a low left-foot finish into Gaston's near post to give the Taysider's some much-needed breathing space.
McAlister's strike was one to savour after the week before his spectacular strike against Kilmarnock in the league had been controversially chopped off.
But their lead was short-lived as Morton hit back within just two minutes.
Peter MacDonald saw his initial shot blocked inside the box by on-loan Celtic ace Toshney.
But Tidser was quickest to react to the loose ball and despite the best efforts of Douglas, he could not keep the ball out as he got a hand to it but could only watch as it trickled into the bottom corner of the net.
There was not much to set pulses racing after the break although McAlister did try to enliven things with an ambitious shot from 30-yards which Gaston did well to hold.
But after 65 minutes they went 2-1 ahead through Toshney's glancing header from McAlister's corner which David O'Brien tried to clear but could not keep it out.
In the 71st minute, Dundee killed the contest when Riley's cross was headed by Scott and Nish was on hand to nod home from eight yards.
Baird made it 4-1 after 74 minutes with a 12-yard finish after Nish's knockdown from Kevin McBride's free-kick.
And Gallagher rounded things off after 82 minutes with a header from McAlister's corner.