The hosts went ahead with a fantastic strike from Danny Haynes in the 19th minute before Jamie Ward equalised for the Rams from the penalty spot midway through the second half.
Charlton were reduced to 10 men in the 71st minute when Michael Morrison picked up a second booking for fouling Richard Keogh in the area, but Derby were unable to snatch a late winner.
Clough said: "We started the second half strongly and we deserved the point. That is two out of three we've conceded on the road and we need a bit of luck.
"We did everything we could - we got it off the line. We certainly didn't want to lose today, we're hardly a negative side.
"Nathan Tyson and Theo Robinson did well up front. We were more solid with Ben Davies on the side It was a bit of a throwback, trying to battle."