Speaking after the game though, former Blues midfielder Lampard - who was making his an emotional return to Stamford Bridge - thought that his new side deserved more from the game than they ultimately got.
Chelsea assistant and Lampard's former team-mate Zola was frustrated with the Blues' defensive efforts, but did concede that much of that was down to Derby's play.
Chelsea took the lead through a Derby own goal by one of the Chelsea on loan youngsters of all people.
West Ham pulled a goal back with a header from substitute Perez but Tottenham's Spanish striker Llorente ended the contest with a fine close-range volley from a corner 15 minutes from time.
Showing some serious fight, however, Derby County quickly equalised after the initial blunder; Jack Marriot finishing the chance well.
When is Chelsea vs Derby County? Tomori and Richard Keogh controlled him better than they could control their own limbs.
Walking in through the front door is an altogether different experience, and one you sense that even Lampard - who has been back to Stamford Bridge many times since his retirement from playing - was a little unprepared for.
Lampard also conducted a post-match lap of the pitch to applaud the Chelsea supporters, who will always count him as one of their own, for their extensive support on the night.
The Premier League giants moved into the EFL Cup quarter-finals after the tight win at Stamford Bridge on Wednesday. I joke because I'm proud.
But Fabregas' goal proved the victor as he took advantage when the ball ricocheted kindly from Chelsea right-back Zappacosta, who was involved in all of their goals. Chelsea won 6-1 on that occasion as the Rams were on they way back down to the Championship.
Chelsea were drawn against Bournemouth and will hist them in the next round of the Carabao Cup.
The Rams nearly forced the game into penalties when David Nugent, who came on as a second-half substitute, thought he had scored but saw his shot rebound off the inside of the post.
"I just hope and pray that they can go there and I think as well it's an opportunity for Chelsea to look at their two players under their own lights and see how they are getting on against top, top opposition".
A stumbling Waghorn passed up a one-on-one chance to complete a quickfire turnaround and Derby punished themselves through more calamitous defending.
Chelsea signed Kovacic on a season-long loan on transfer deadline day to bolster their midfield options.
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.
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