DF David Luiz, 6 - Brought on for Christensen to give Chelsea greater defensive solidity, he broadly did his job as Derby pushed in vain for an equaliser in the final 20 minutes.
MF Cesc Fabregas 6 - His long-range passing was off and he was caught out for Derby's first goal, but timed a run into the penalty area perfectly to score Chelsea's crucial third goal with their first shot on target. But now they have gone out and they can concentrate on the Championship, which is what is so important now.
The quarter-final draw pits Arsenal against Spurs, while Chelsea take on Bournemouth and Middlesbrough play Burton.
The first was from the Rams' on-loan Chelsea defender Fikayo Tomori, who played against his parent club along with Mason Mount, and the second by Richard Keogh.
Despite the defeat that knocked them out of the competition, Lampard and his Derby County side can keep their head up high and brimming with pride as they put in a performance that was the flawless testament to their ambition under the new manager.
Chelsea began the hospitality early by kindly granting permission for Mount and Tomori to play against them.
But him and Cahill looked awkward on the ball and assistant manager Zola admitted that the duo struggled. "I'm a lucky man to have a night like that and I feel it".
After twice equalising, the second time through Waghorn, Frank Lampard's side forced a number of good chances, notably through Jack Marriott and David Nugent, but were unfortunately denied. "Derby were excellent. They created us more problems".
Head coach Maurizio Sarri said after that match that Hazard's fitness remains of critical importance, and Zola said that he is on track to return to the team when Crystal Palace visit Stamford Bridge on Sunday. Who goes through clear against Chelsea like we did on goal?
It will be the Blues' third match in the space of six days, having beaten BATE Borisov and Burnley in the Europa League and Premier League respectively.
Chelsea are still unbeaten in competitive games this season.
Marriott has been directly involved in six goals in his past five games for Derby in all competitions (four goals, two assists).
But the visitors played a full role in a frantic first half, looking sharp as they came forward and were back on level terms before Fabregas scored what turned out to be the decisive goal.