Gavin Rae steered a header home four minutes into the second period of extra-time and Scott Vernon tapped in from close range five minutes later to put the game out of Morton's reach.
The Irn-Bru First Division side were more than a match for their Clydesdale Bank Premier League opponents and can consider themselves unlucky not to have at least forced penalties, having enjoyed the better chances during the opening 90 minutes.
Torrential rain in the hours leading up to kick-off left the Cappielow surface sodden and both teams struggled to adapt to the difficult conditions in the early stages.
When the first chance did arrive it fell to the hosts, Stephen Stirling's curling 25-yard effort forcing Jamie Langfield into an excellent stop.
Aberdeen were struggling to gain a foothold in the match and when they did eventually manage to create a positive breakaway, David O'Brien rashly lunged in on Isaiah Osbourne - earning himself the first yellow card of the match.
Ryan Jack stung the palms of Derek Gaston with a shot from distance after 16 minutes as the SPL side began to stamp their authority on the game and they passed up a glorious opportunity to take the lead four minutes later.
Gaston parried Ryan Fraser's cross directly to Rae, but the midfielder could only manage a weak effort straight into the hands of the grounded keeper.
Peter Weatherson dragged an effort wide of Langfield's post as Morton continued to try to attack whenever possible.
It was the visitors who continued to look more threatening however and only Stirling's brave block prevented Fraser from breaking the deadlock with a fizzing half-volley.
Two minutes before half-time Aberdeen were dealt a blow when Stephen Hughes limped off and was replaced by Chris Clark.
The hosts made a bright start to the second half and O'Brien almost profited after dispossessing Jack on the edge of his own penalty box, but his low strike could only find the side-netting.
On the hour Alan Moore made his first change of the evening, with Wallace replacing Stirling.
It took Aberdeen 17 minutes to create their first chance of e second half and after some good build-up play by Jonny Hayes, Rae slid the ball wide of the target.
Vernon was struggling to make an impact on the game and he should have done much better when met Jack's cross, but the ball sailed harmlessly wide of Gaston's goal.
The Morton keeper was called into action a minute later when he produced an excellent two-handed stop from Rae's long-range drive.
A poorly-timed lunge on Hayes earned McLaughlin a yellow card with 19 minutes remaining, before the hosts were forced into a second switch as Weatherson hobbled off to be replaced by Archie Campbell.
Some heroic defending by McLaughlin prevented Rae breaking the deadlock following a dangerous corner from Hayes after 74 minutes.
The hosts almost grabbed the opener when a quick breakaway released Campbell and only a wonderful save by Langfield prevented the substitute scoring.
Reid became the third Morton player to be booked when he hacked down Hayes whose resulting free-kick clipped the bar.
Fraser was withdrawn for Peter Pawlett with four minutes remaining.
Aberdeen looked the much fresher of the two sides as extra-time began and Dyer was forced to turn Pawlett's cross behind under pressure from Vernon.
A superb flowing Morton move ended with O'Brien squaring the ball across goal but nobody was on hand to finish it off.
David Graham's shot was spilled by Langfield in the opening minute of the second period of extra-time, but the home side were dealt a cruel blow three minutes later.
Hayes produced a pinpoint delivery from the right and Rae headed into the bottom right corner.
Morton reacted by throwing men forward and they almost paid the price when Vernon led a quick break before unleashing a shot which was well stopped by Gaston.
With six minutes remaining Hayes released Rae and the midfielder unselfishly rolled the ball to Vernon to slot into the gaping goal.