From then on the visitors looked comfortable as Morton never really threatened a shock.
Celtic will now go away to Irn-Bru Second Division side Stenhousemuir in the next round, if Dunfermline, who were thrown out of the competition by the Scottish Football Association for breaking various rules in their win against the Warriors, do not win their appeal.
Cappielow enjoyed a near full-house of 10,191 for the visit of the Parkhead men who travelled with the ropey record of one win in their previous fives games.
First Division Morton started with impressive and inevitable enthusiasm but it always looked like being a difficult night.
Allan Jenkins fired wide from 20 yards after the home side had fashioned a half-chance in the fifth minute and moments later there were unsuccessful cries for a penalty when Celtic captain Darren O'Dea appeared to tug at Brian Wake in the box.
Celtic slowly asserted themselves and were soon making inroads towards Stewart who dealt easily, however, with a couple of efforts from McGinn.
Then in the 19th minute, after slack play by defender Glenn Loovens in giving the ball away to Peter Weatherson, Celtic goalkeeper Artur Boruc had to make a decent save from the Morton striker's long-range drive.
An open game swung from end to end with Fortune failing to control a Lee Naylor cut-back from the by-line in a penetrating Celtic attack, just before Aiden McGeady fired wide after a driving run into the home box.
Celtic had the edge in class and in the 35th minute it showed when McGinn fired them ahead with a perfectly-executed goal.
As Celtic probed down the left, the Northern Ireland international was perfectly placed to meet a Fortune cut-back and thrash a left-footed shot past Stewart from 12 yards.
Moments later McGinn tried to return the favour when he fired across the Morton goal from wide on the right and a stretching Fortune turned the ball wide with the net unprotected.
Celtic were well in control when referee Craig Thomson blew the half-time whistle and the Morton players must have trooped off wondering how they could turn the tie around.
The early stages of the second half did not suggest anything untoward would happen to the visitors, who beat a path towards the Greenock goal, if with less of a cutting edge than boss Tony Mowbray desired.
In the 58th minute Stewart was called into action for the first time in the second half when he dived to his right to parry a powerful Georgios Samaras drive away for a corner, which Morton survived.
In the 71st minute, as the home side broke into the Celtic box for the first time in the second half, only a great blocking tackle by Lee Naylor on Brian Wake prevented the Morton striker getting in a shot from close range.
Six minutes later Samaras bundled the ball into the net at the back post from a Naylor corner which had been flicked on by O'Dea, but Thomson ruled the goal out for reasons which were not immediately clear.
With four minutes remaining Naylor ran on to a Samaras knock-down and crashed a wonderful 30-yard volley off the post, moments before Fortune failed to control the ball a few yards from goal to pass up on a great chance to clinch the tie.
But the game fizzled out to conclusion with no expected rally from the home side.