Eden Hazard profited from a mistake by Rams captain Richard Keogh to coolly slot the visitors in front halfway through a first half dominated by Jose Mourinho’s men, before Filipe Luis doubled the lead with a stunning free kick.
Craig Bryson pulled one back with a fine finish 20 minutes from time, before the hosts were reduced to 10 men as last man Jake Buxton was sent off for a foul on Loic Remy, and Andre Schurrle made them pay as he fired in a third to seal victory.
There were two blows for the visitors as Kurt Zouma was stretchered off in first-half stoppage time after being knocked unconscious as he went up for a cross with goalkeeper Petr Cech, before Didier Drogba limped off after slipping on the rain-hit surface.
Derby had to cope without Ryan Shotton after the defender was sent off in their Championship defeat at Middlesbrough, so Buxton was introduced as Will Hughes replaced Jeff Hendrick in midfield.
The visitors, meanwhile, rang the changes but Jose Mourinho still named a strong line-up for the trip to the iPro Stadium. Schurrle and Cesc Fabregas came back into the XI as Drogba was given the task of leading the line.
It was a quiet opening spell, only marked by Schurrle’s early booking for a foul on Johnny Russell, but Chelsea dominated possession without looking in a hurry to score.
But the visitors took the lead on 23 minutes after excellent pressing forced a mistake from skipper Keogh, who surrendered possession to Drogba. The striker found Fabregas who quickly released Hazard, and the Belgian worked some space beautifully before firing a low shot beyond the goalkeeper.
Luis tested Lee Grant moments later, firing a stinging volley at the Derby goalkeeper after superb work by Fabregas in midfield.
The half ended on a sour note for the visitors as Cech punched away a corner and caught Kurt Zouma in the process. The defender was knocked unconscious by the goalkeeper’s shoulder and, after several minutes of treatment, the Chelsea man was stretchered off.
Derby started the second half brightly, and Hughes nearly levelled things as he tested Cech with a low effort after excellent work by Johnny Russell down the right-hand side.
But Chelsea quickly added to their lead as Drogba was brought down by Hughes in midfield, and Luis stepped up to fire in a stunning 30-yard free kick, which rifled into the top corner and gave Grant no chance.
Drogba then had to go off as Chelsea had to make a second substitution because of injury, after the forward went down awkwardly on the rain-soaked pitch.
The hosts were then given a lifeline as substitute Branislav Ivanovic gave the ball away and Bryson played a one-two with Russell before firing a 20-yard effort past Cech and in off the post.
It wasn’t to be for Derby, though, as Buxton was sent off in bizarre circumstances. Grant slid out to take the ball away from Remy and looked to have handled outside the box, but after consulting with his assistant, referee Jon Moss gave the defender his marching orders for a tug on the striker.
Chelsea sealed the victory soon after as Fabregas played in Remy, who drew a fine save from Grant, but the goalkeeper could only parry it into the path of Schurrle and the German made no mistake to fire the ball low into the net and confirm the visitors’ passage into the last four.
GLENN HODDLE’S VERDICT
“They’re a class act with class players – Cesc Fabregas being one. Eden Hazard as well excelled today. They press the ball hard and they work hard, and then when they’ve got the ball they’ve got a lovely balance to their play. It was a gallant effort from Derby and they’ll take the experience away from this – they’ll be hoping they’ll be playing against this sort of quality every week next year.”