The Clarets were without a win in five games heading into the fixture, but produced a solid victory courtesy of goals in each half from Charlie Austin and Michael Duff.
The Rams failed to produce a response but Clough insisted he was satisfied with their application.
He said: "I'm not too disappointed with the performance, it's the goals that we are conceding at the moment, both were technically own goals as well.
"First half we weren't too good in terms of creating but we certainly restricted Burnley to only a few chances, but there is not a lot you can do when someone is having a shot that is going two yards wide and it deflects into the bottom corner.
"The second one we should have defended better and it was a period of the game when we were on top and pushing for an equaliser.
"We moved Michael Jacobs up with him (Conor Sammon) at the start of the second half and played 4-4-2 which our supporters didn't realise but the team did so that's what mattered.
"We were pushing for the equaliser and we had a chance but it was a good save by the keeper. We hit the post from the rebound and a couple of minutes later it is 2-0 and it's almost game over."