It was Derby's first League Cup win since their run to the 2008/9 semi-finals and a hard-earned result for Clough's Championshipside against League One opponents.
Derby had chances to increase their lead, but the best was headed off target by an unmarked Ben Davies.
"This has been a long time coming and it would be nice if it was the start of a cup run," said Rams boss Clough.
"The League Cup tends to be a competition for the underdogs and you come to places like this knowing they will make it very difficult for you.
"We were up against a good side in Oldham, who are a credit to their manager (Lee Johnson). We've come out with a victory and a place in the next round, so I'm perfectly happy with that.
"I wasn't happy with our first-half performance, but we controlled it after half-time and could have scored more than one - we should have had at least one penalty, maybe three.
"The goal we did score was brilliant. There was a great ball from John Eustace and Conor Sammon rolled it across for Michael Jacobs, so it was a great team goal. The clean sheet was pleasing as well. Our centre-backs and 'keeper deserved a clean sheet against Blackburn on Sunday but they got that reward here."