Eden Hazard's ninth goal of the season put the London club in front at half-time and Filipe Luis opened his Chelsea account with a sublime free-kick early in the second half to double their lead.
Craig Bryson gave Championship high-flyers Derby hope when he pulled a goal back 19 minutes from time at Pride Park on Tuesday, but when Jake Buxton was sent off for a professional foul on Loic Remy a comeback looked highly unlikely.
Andre Schurrle gave the Premier League leaders breathing space when he added a third goal and there was no way back for Steve McClaren's side.
The Chelsea manager told Sky Sports: "Derby are a Championship side but not a typical Championship side, they have more quality than that and they play good football with fantastic support in their stadium.
"We played in a very professional way and controlled the game for most of the time. They fought to the end and they had a good period after they scored when they were able to put us in trouble."
He added: "Cesc Fabregas is playing so well for us. What I like is not only is he a great player, but as a person he is a professional.
"He is a world champion, he comes here to play in the Capital One Cup and plays like that. That is the character I like and what I want to see from all of my players."
