The Rams were the better side for the majority of the match, particularly in the first half, and were holding on relatively comfortably when the Foxes improved after the break.
Chris Martin's 42nd-minute goal looked like it might give Derby a first triumph on Leicester soil since a 3-0 success at Filbert Street in February 2002 - when this was a Premier League fixture - until the introduction of Anthony Knockaert.
The Foxes substitute equalised with a stunning 25-yard free-kick after 78 minutes and then had a hand in Danny Drinkwater's winning goal just three minutes later.
Clough felt his team were architects of their own demise after conceding needless free-kicks.
"We felt we'd done enough to win the game throughout the 90 minutes, certainly in the first half," said Clough.
"It was an excellent away performance and the one-goal lead was not sufficient reward for how we played.
"I felt we deserved more than that lead, and we had a couple of penalty appeals we felt we should have got as well.
"With 10-15 minutes to go we are relatively comfortable and I don't think Lee Grant has had anything to do during the game up to that point.
"They were both needless free-kicks to give away so we feel we've contributed to our own downfall there."
Clough's side had a couple of penalty appeals midway through the first half, but both were waved away by referee Dean Whitestone.
First Craig Bryson tumbled over Leicester debutant Marcin Wasilewski and then Paul Coutts went down under a challenge by Ignasi Miquel.
The Derby midfielder was penalised for handball in the second instance but more worryingly for the Rams he lay in some pain on the ground before having to be helped from the field and replaced by Ben Davies.
Coutts' injury looks set to rule him out of Derby's trip to rivals Nottingham Forest on Saturday.
Clough said: "He's very sore at the moment, although he was more angry about the fact he hadn't been given a penalty rather than his knee.
"We will get him scanned on Wednesday but it's not looking too favourable at the moment. I think he'll definitely be out on Saturday but I don't know how long beyond that just yet."
Leicester boss Nigel Pearson, who made eight changes to his starting line-up, hailed Knockaert's influence after his 63rd-minute introduction from the bench.
"We know he's a talented player," said Pearson.
"I can't remember too many shots he's had like that hitting the target, so well done to him.
"He still has a lot to learn but he's come on leaps and bounds in the 14 months we've had him.
"It's good to get through, that was the aim at the start of the night.
"The performance was a mixed bag, but that happens when you make so many changes. But I expected that lack of fluency a little, although I wanted us to show a little more impetus.
"Although we were 1-0 down I don't think Derby caused us too many problems, the two centre-halves did well."