Goals from Theo Robinson and Jamie Ward either side of Chris Eagles' strike put the Rams in front at half-time.
But Burnley fought back and Eagles levelled for a second time on the rebound after his penalty was saved by Brad Jones following Ward's sending off for a professional foul
Eagles then provided the cross for Wade Elliott to head Burnley in front and Chris McCann, starting his first game of the season following reconstructive knee ligament surgery, sealed the win midway through the second half.
Clough said: "We were extremely pleased at half-time with the performance, but disappointed not to be more than 2-1 ahead we created enough chances to be more than one goal in front at half-time.
"They came out in the second half and started strongly and at that point you've got to hang on in there which we were doing, but then we conceded a penalty and were down to 10 men which made it very difficult in the heat after that."
Source: PA...
Source: PA