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.
Chelsea are still in the hunt for four trophies and Mourinho was pleased with the way his side approached the tie against a Derby side that he knew were capable of causing them problems.
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."
Mourinho also picked out Cesc Fabregas for praise following another impressive display from the Spain midfielder.
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."
Mourinho revealed that Kurt Zouma will be available to face Stoke City in the Premier League next Monday after being carried off with a cut to his mouth following a collision with Petr Cech.
He also added that striker Didier Drogba has a swollen ankle after slipping on the wet turf, but is not seriously injured.
