The visitors took the lead after 15 minutes when Kevin McBride was unfairly challenged in the box. Ryan Conroy clinically dispatched the spot kick.
Morton pressed hard for long spells afterwards and were aggrieved that they too were not awarded a penalty when David O’Brien tumbled as Dundee goalkeeper Rab Douglas blocked his progress. The winger was also unfortunate as Andy Jackson’s cross just eluded his outstretched boot with an open goal beckoning.
After the break, the hosts were left deflated when Conroy notched up his second with a delightful lob.
Shortly afterwards, the hosts were reduced to ten men as Peter Weatherson picked up two yellow cards in quick succession, enabling Dundee to see out the game with something to spare.