It was a controversial award, with Mario Campagna penalised for a push in the box on Gordon Pope as he tried to get on the end of a Ryan McCord fee-kick which the keeper collected.
Scott Paterson did the needful from the spot, sending Grant the wrong way.
Overall United deserved their win because they had more and the clearer-cut chances then their hosts.
Indeed, apart from former United youth player Ryan O'Hara forcing John Gibson to tip his long-range free-kick on to the post on 18-minutes, Gretna's youngsters rarely threatened.
Before the break, Grant twice denied Ryan McCord as well as Fraser Milligan.
After the visitors had gone ahead, Ross McCord could have made it two when he shot past from a good position, but in the end United had little difficulty hanging on for the three points.
Gretna: Grant, Campagna, Poulson, Shultz-Eklund, Faulds, Sharples, Robinson (Palmer), O'Hara, Cusak (Taylor), McKinlay, Hughes (Smee). Subs not used: Pattison, Johnson, Moffet.
Dundee United: Gibson, Milligan, Fleming, McGrath, Pope, Anderson, Brophy, Ross McCord (Mathieson 70), Love (Smith 60), Ryan McCord, Paterson. Subs not used: Colgan, Travers, Forrest.