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 two clubs, divided by the 14 miles (22 kms) of the Brian Clough Way, are only a point apart in the lower regions of the Championship.
Derby, showing no ill-effects from their Old Trafford exertions of three days ago when they lost to Manchester United in the second leg of their League Cup semi-final, were on top from the start.
They took a deserved lead after 36 minutes with a goal started and finished by Rob Hulse. The lively forward was on hand to sweep in a Paul Connolly cross after a Paul Green dummy wrong-footed the visiting defence.
Forest came back strongly after the break and levelled in the 64th minute with a similar goal. This time it was Nathan Tyson supplying the dummy and former Derby striker Rob Earnshaw turning neatly to finish from 10 metres.
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 replay at the City Ground will take place on February 4.
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.
