Charlie Austin's 23rd goal of the season set up the hosts' victory, before Michael Duff's 74th-minute deflected header wrapped up the points.
The defeat means the Rams have now lost five of their last six away games, and Clough concedes his team needs to improve.
He said: "I'm not too disappointed with the performance, it's the goals that we are conceding at the moment.
"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.
Clough continued: "We moved Michael Jacobs up with him (Conor Sammon) at the start of the second half and played 4-4-2.
"We were pushing for the equaliser and we had a chance but it was a good save by the keeper but we hit the post from the rebound and a couple of minutes later it is 2-0 and it's almost game over."