The Premier League title hopefuls needed to ride their luck against their Championship opponents, though, as they benefited from two own goals during a remarkable first half and then saw a late David Nugent shot hit the inside of the post.
Derby held their own for long spells and quickly hit back on the first two occasions that they fell behind, with goals from Jack Marriott and Martyn Waghorn cancelling out own goals from Fikayo Tomori and Richard Keogh for the luckless visitors.
There was an element of controversy about Chelsea's winner too as Cesc Fabregas finally provided a home contribution to the scoreline shortly before half time, and Derby could not produce a third comeback as the Blues progressed into the quarter-finals at the expense of their record goalscorer.
Chelsea took the unusual step of allowing Derby to select on-loan duo Tomori and Mason Mount against their parent club in the hope that the experience would aid their development, but there was an unexpected short-term benefit too when Tomori inadvertently gave the hosts the lead in calamitous fashion after only five minutes.
Davide Zappacosta's low cross from the right flank should have been comfortably dealt with by the youngster, but he completely missed his attempted clearance and the ball bounced off his standing leg before dribbling past Scott Carson.
Chelsea smelt blood early on and went pushing for a quickfire second as Willian dragged an effort wide, but Derby responded well and were level within just four minutes of going behind.
Derby should have taken the lead after 17 minutes when Marriott turned provider by clipping a pass through for Waghorn, whose need to keep the ball on his left foot led to him stumbling and poking a tame effort straight at the keeper.
It was a golden opportunity for the Championship outfit to turn the game around, and it looked as though it could prove costly when they incredibly fell behind to a second own goal just four minutes later.
Zappacosta was again the architect with another low cross into the box, but Keogh also should have done better as his attempt to put the ball behind only saw him join his centre-back partner on the scoresheet for the wrong team.
Once more Chelsea quickly went in search of more goals after taking the lead, with the ever-dangerous Ruben Loftus-Cheek dragging one solo effort wide before Willian flashed a 20-yard shot narrowly over the crossbar.
This time the lead lasted just six minutes, though, as once again Derby responded quickly. Chelsea loanee Mount was involved as he latched on to a through-ball down the left channel before threading a low pass to the back post, where Waghorn was waiting to apply a simple finish and atone for his earlier miss.
Despite having two goals to their name at the time, it wasn't until the 38th minute that Chelsea actually registered their first shot on target when Emerson Palmieri - who scored against Liverpool in the third round - saw his weak effort comfortably saved by Carson.
After a bizarre but brilliant opening 40 minutes, some normal order was finally restored when Chelsea scored a goal via their own means, although it wasn't without controversy as Derby felt that Emerson should have been penalised for a barge on Tom Lawrence in the build-up.
Enough time elapsed before the goal for Derby to deal with the danger, but they failed to do so as Fabregas applied the finishing touch by sweeping his first-time shot beyond Carson just four minutes before half time.
This time Chelsea were able to hold on to their advantage as they went into half time ahead, and they emerged for the second half with a greater degree of control over the game.
It took until the hour mark for the first clear chance of the second half, with Cahill escaping the attentions of Keogh but putting his downward header too close to Carson.
Derby soon began to cause Chelsea problems again, though, with Marriott stinging the palms of Caballero with a powerful drive and Mount seeing a shot deflected narrowly wide going into the final 20 minutes.
Alvaro Morata could have made things more comfortable for the hosts with a trio of headed chances in as many minutes, the best of which saw him draw a smart reaction save out of Carson having darted to the near post to meet a corner.
The visitors really began to ramp up the pressure in the final five minutes, and it was Caballero who suddenly found himself busy with saves to deny Keogh - who just could not get enough on his finish after expertly taking the ball down inside the area - and Mount in quick succession.
Caballero then kept out Marriott at the near post, and when the keeper was beaten in the 89th minute, the woodwork came to his rescue as Nugent's measured finish came off the inside of the post and bounced straight back into the arms of Caballero.
It was the type of luck which had run for Chelsea and against Derby all night, and in the end the Rams were left with nothing to show for their spirited efforts as their EFL Cup journey - which had already seen them claim the scalp of Manchester United - ended in the fourth round.
Chelsea, meanwhile, are now unbeaten in 15 matches across all competitions as they continue to challenge on all fronts under Maurizio Sarri.
CHELSEA (4-3-3): Caballero; Zappacosta (Azpilicueta 78'), Cahill, Christensen (Luiz 66'), Emerson; Kante, Fabregas, Kovacic; Loftus-Cheek (Pedro 69'), Morata, Willian
DERBY (4-5-1): Carson; Bogle, Keogh, Tomori, Malone (Forsyth 82'); Waghorn (Bennett 62'), Wilson (Nugent 82'), Huddlestone, Mount, Lawrence; Marriott