Derby County 1 Chelsea 3: Premier League leaders progress
Eden Hazard's ninth goal of the season gave the Premier League leaders a half-time lead following a mistake from Derby captain Richard Keogh.
Filipe Luis then doubled Chelsea's lead with a sublime free-kick, his first goal for the club, early in the second half and just after Will Hughes had spurned a great chance to equalise.
Craig Bryson pulled a goal back with 19 minutes remaining to give Steve McClaren's side hope, but Jake Buxton's sending off for a professional foul on Loic Remy made a comeback unlikely.
Andre Schurrle sealed the victory with a third goal and Chelsea remain in the hunt for four trophies, but their triumph could have come at a cost as Kurt Zouma and Didier Drogba both sustained injuries.
Drogba replaced Diego Costa in a strong Chelsea starting line-up, with Zouma coming in for the rested Gary Cahill and Cesc Fabregas back following a suspension.
Derby were without Ryan Shotton due to a ban, so Buxton started and Hughes was preferred to Jeff Hendrick in midfield.
Chelsea dominated possession in the opening stages, but the home side worked hard off the ball to frustrate them and were growing in confidence before they fell behind following a mistake from Keogh 22 minutes in.
The Derby skipper was put under pressure on the edge of his own penalty area and Fabregas latched onto the loose ball to feed Hazard, who darted inside the box and took his time before beating Lee Grant with a clinical finish.
Filipe Luis then stung the palms of Grant with a venomous left-footed drive before Keogh's fine tackle thwarted Drogba as he looked destined to find the back of the net.
Zouma's game was cut short just before half-time after he collided with Petr Cech from a Derby corner and was carried off following a lengthy period of treatment, with Branislav Ivanovic replacing him.
Derby finally broke Chelsea down five minutes after the break, when the lively Johnny Russell skipped past John Terry on the right touchline and cut the ball back for the onrushing Hughes, but the England Under-21 international was denied by Cech.
Filipe Luis then gave the visitors breathing space soon after that missed opportunity, giving Grant no chance with a stunning free-kick from 25 yards out to open his Chelsea account in style.
Chelsea suffered another injury blow when Drogba limped off with an ankle injury after slipping, so on came Loic Remy.
The tie appeared to be over, but Bryson provided the quality Derby had been lacking in the final third when his right-footed finish went in off the far post after Russell had picked him out on the edge of the box in the 71st minute.
Derby were dealt a major blow 12 minutes from time, though, as Buxton was dismissed for pulling back Remy when it looked as though Grant would be the man in trouble when he slid out of his area with the ball in his grasp.
And Schurrle ended the contest when he tucked the loose ball home after Grant had kept out Remy's effort after 82 minutes.