A mistake by Dundee skipper Barry Smith gifted Gretna an opening midway through the first half. He failed to deal with a seemingly innocuous long ball allowing James Grady to get in behind him and send a dangerous cross into the box.
However, Stuart McCluskey got back to make a fine challenge just as Gretna's top scorer Kenny Deuchar was about to pull the trigger.
The second division champions saw more of the ball as half-time approached and Gavin Skelton had a deflected effort well held by Dundee's Kelvin Jack.
Then, in first-half injury-time, Gretna took the lead. Deuchar capitalised on a ball that flicked off the head of Dundee defender McCluskey before rounding the keeper and rolling a shot into the empty net.
Deuchar could have doubled Gretna's lead at the start of the second half but headed Skelton's cross wide.
Gretna scored their second goal in controversial circumstances in the 57th minute. The referee awarded a penalty after Ryan McGuffie ran in to Dundee defender Paul Dixon. McGuffie took the spot kick and sent the keeper the wrong way.
Dundee's woeful afternoon was rounded off after 81 minutes when Smith stuck out a leg to deflect the ball into his own net.