Derby, who are coached by Chelsea legend Lampard, went behind when Tomori inadvertently deflected a cross into his own net.
Martyn Waghorn drew the visitors level for a second time before Cesc Fabregas struck what proved to be the victor in the 41st-minute. Leicester or Southampton will play Manchester City or Fulham.
"He is certainly much better", Zola told his pre-match media conference on Tuesday ahead of their League Cup clash against Derby County.
Gianfranco Zola afterwards played down the notion that Chelsea's edge had been blunted by playing in a stadium chanting for the opposition manager, instead crediting Lampard's game plan of blocking passes into Cesc Fabregas and forcing those behind him into uncomfortable decisions.
It will not be an easy task however, for the Rams to grab a second Premier League scalp given Chelsea's impressive start to the season under Maurizio Sarri.
The size of the challenge to beat Chelsea was huge for Derby, with no win here since 1983 and no clean sheet since Peter Shilton's blank in 1987.
Hazard, who scored a sensational victor against Liverpool in the last round, has missed each of Chelsea's last two games against BATE Borisov and Burnley as a result of a back injury thought to have been suffered after some rough treatment at the hands of Manchester United.
"Gary Cahillplayed for the first time in a while".
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Why are Mason Mount and Fikayo Tomori allowed to play?
Predicted Derby County XI: Carson; Bogle, Tomori, Keogh, Forsyth; Wilson, Huddlestone; Waghorn, Mount, Lawrence; Nugent. Buoyed by his side's win on penalties at Old Trafford against his former Chelsea boss Jose Mourinho in the previous round, he was dismayed to witness the errors from his side, which made their task all the harder.
Attention now turns back to league action this weekend and the manager wants his side to continue the good form built up. He showed frustration at the goals his team conceded but, true to his word, declined to celebrate the ones they scored, settling instead for a silent, satisfied swig of water.
Ruben Loftus-Cheek is now enjoying the best goalscoring spell of his career, netting in two consecutive games for the first time.
27' - GOAL! Chelsea 2-2 Derby: They have only gone and equalised again!
It was a Lampard-esque goal and Derby's protests were dismissed by Moss.
Key Stat: Chelsea (31) and Derby (30) have combined to score 61 goals in all competitions this season.