Second-half strikes by Jan Zemlik and Gavin Swankie booked Dundee's passage into round two of the CIS Insurance Cup at the expense of Morton.
The Dark Blues ought to have had things wrapped up by half-time, but Czech striker Zemlik deflected in a Scott Robertson piledriver on 47 minutes.
Robertson sent Swankie clear five minutes later and the midfielder scored from an angle at the second attempt.
Dundee should have added to their tally but were still comfortable winners.
Dundee: Roy, Worrell, Griffin, Palenik, Dixon, Daquin (O'Brien 62), McDonald, Robertson, Swankie, Zemlik (Lyle 76), Sturm.
Subs Not Used: McHale, Davidson, Samson.
Goals: Zemlik 47, Swankie 52.
Morton: Robinson, Weatherson, Walker, Harding (Graham 76), Greacen, Finlayson (Shields 62), Millar (Keenan 54), Stevenson, Templeman, McAnespie, McAlistair.
Subs Not Used: Coakley, McGurn.
Ref: E Smith (Scotland).