It was a disappointing afternoon for the Taysiders, who will now concentrate solely on their chase for the First Division title, while Gordon Chisholm's men can bask in their achievement of reaching the semis.
Dundee boss Alex Rae named himself on the bench amid a selection crisis.
Rae was without seven first-team players through injury, suspension and ineligibility and brought in the likes of strikers Bob Davidson and Jan Zemlik, while on-loan Hibernian midfielder Patrick Noubissie made his debut.
The Doonhamers, meanwhile, were boosted by the return of defenders Robert Harris and Jim Thomson, who returned from injury and suspension respectively.
The home side were looking to reach their first semi-final for more than fifty years as well as extending an unbeaten run to 10 matches.
Dundee winger Freddie Daquin looped a header over, but it was the home side who had the better of the early stages.
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