Leigh Griffiths fired the only goal of the game as Dundee secured their first away win of the season at Cappielow.
Morton almost broke the deadlock when Tony Bullock parried Peter Weatherson's effort and Jim McAlister's follow up was blocked on the line by David Cowan.
Bullock then turned a Weatherson shot onto the post and Colin McMenamin had a goal chalked off for offside.
Griffiths scored the winner 13 minutes from time when his 15-yard shot sailed over Morton keeper Colin Stewart.
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