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.
It was all Dundee and midway through the first-half but they had to wait until the 29th minute to take the lead through McAlister.
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.
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.
Gary Deegan scored his first ever goal for Hibernian as the Irishman's spectacular strike secured a win over Aberdeen which carried them into the quarter-finals of the Scottish Cup.
However, it was Easter Road goalkeeper Ben Williams who was the true hero for the hosts, producing five excellent second-half saves as well as plunging to parry away a late Scott Vernon penalty.
After a turgid first period, Hibs took control of the contest immediately after the interval, with Deegan producing a dipping swerving drive from distance to break the deadlock.
Aberdeen, roared on by a vociferous following from the north, produced a late rally, with Russell Anderson, Scott Vernon, Niall McGinn and Peter Pawlett all coming close, but Williams was in incredible form.
Even better was to follow from the big Englishman as he saved Vernon's spot-kick with six minutes to play - his fourth penalty save this season - after Jonathan Hayes had been clumsily felled by Hibs skipper James McPake.