A Mark Farren goal at the Brandywell gave Derry their first Eircom Premier Division win in six games.
City dominated the first half before striker Farren smashed home at the near post from Eddie McCallion's cross five minutes after the break.
Derry keeper Gerard Doherty made a great save to keep out John Kearney's close-range strike on 67 minutes.
Cobh's John Meade was red-carded two minutes later when he brought down Farren as he went clear on goal.
Third-placed Derry are now six points behind leaders St Pat's, who drew 2-2 away against champions Drogheda United.
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