Maurizio Sarri's men were handed an early lead through Fikayo Tomori's unfortunate own goal before Jack Marriott restored parity for the visitors shortly thereafter.
Derby put in an excellent display at Stamford Bridge, twice battling back from a goal behind in the first half before Cesc Fabregas grabbed what turned out to be the victor just before half-time.
This fourth-round clash provided an opportunity to do so and Lampard was serenaded throughout by Chelsea supporters, and he responded by acknowledging them and the Derby fans at the end.
Hazard has made a brilliant start to the Premier League season following a return of seven goals in nine games in the English top flight.
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. The Premier League is a different level.
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.
Derby: Carson 6, Bogle 6, Keogh 5, Tomori 5, Malone 6, Huddlestone 8, Mount 7, Wilson 7, Waghorn 6, Marriott 7.
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On Tuesday, Burton Albion and Bournemouth booked their places in the quarter-finals while Premier League leaders Manchester City play Fulham on Thursday.
Zola was also questioned about the recent comparisons between current Chelsea midfielder Ross Barkley and the clubs record-scorer Lampard.
Controversy surrounded what turned out to be the winning goal for Chelsea.
And it is this game from which Derby can gain confidence, with the Blues switching off defensively at times.
"It happens. If he hadn't touched that ball, [Alvaro] Morata would have scored a goal". "So we didn't have control of the game".
Ruben Loftus-Cheek, another player Southgate was watching, tumbled over the advertising hoardings after a shoulder-to-shoulder with Scott Malone and that might have subconsciously influenced referee Jon Moss in not awarding Derby a free-kick when Zappacosta and Tom Lawrence had a similar physical challenge on the touchline.
Derby did well to break down United in historic fashion during the third round, but it will take a lot for Lampard to continue his fairytale story during his upcoming homecoming.