New Derby boss Clough is a former Forest player and his father Brian was in charge of both clubs during their glory years in the 1970s.
There was another old boy connection as Forest are now managed by former Derby boss Billy Davies, who was absent on Friday because of a family illness. Both managers took over their respective clubs earlier in January.
The chief talking point from the latter stages was the broken or fractured wrist suffered by Forest's Paul Anderson as he fell awkwardly.
"The fact that it was the FA Cup made it that bit more special," Clough told his club's website. "It was a bit more competitive and that's what it should be like."
The fourth round continues over the weekend with Manchester United playing Tottenham Hotspur on Saturday and Liverpool facing neighbours Everton in the pick of Sunday's games.
