Eden Hazard and Filipe Luis, with his first goal for the club, put Chelsea in control on a rain-soaked night at the iPro Stadium, before Craig Bryson replied for Derby in the 71st minute.
But the hosts’ hopes of a comeback were extinguished when centre-back Jake Buxton was sent off for impeding substitute Loic Remy, and Andre Schuerrle made the game safe with eight minutes to play.
“Derby are a Championship side, but not a typical one. They have more quality than that and they play good football with fantastic support,” Chelsea manager Jose Mourinho told Sky Sports.
“We played in a very professional way and controlled the game for most of the time. They fought to the end and for a few minutes after they scored they were able to put us in trouble.”
Remy was only on the pitch because veteran striker Drogba had been forced off by an ankle injury earlier in the second half.
French centre-back Zouma, meanwhile, was stretchered off in first-half injury time after colliding heavily with his own goalkeeper, Petr Cech. Branislav Ivanovic replaced him.
Mourinho won the League Cup twice in his first stint as Chelsea manager and he named a strong side against Championship high-fliers Derby, with Hazard, John Terry, Nemanja Matic and Cesc Fabregas all in the team.
The visitors made the breakthrough in the 23rd minute after Hazard blocked an attempted pass from Derby captain Richard Keogh on the edge of the hosts’ penalty area.
– Blades cut down Southampton –
Fabregas steered the ball back into Hazard’s path and the Belgian winger drifted infield before drilling a low shot into the bottom-left corner.
After Cech had saved well from Derby midfielder Will Hughes, Luis doubled Chelsea’s lead in the 56th minute by arcing a fine free-kick over the wall and into the net from 25 yards.
Fate appeared to be conspiring against Chelsea after Drogba hobbled off and Bryson swept home Johnny Russell’s lay-off from outside the box to give former England manager Steve McClaren’s side hope.
But after Buxton had been contentiously dismissed for wrestling with Remy as the Frenchman bore down on goal, Schuerrle completed the scoring by following in after Remy’s shot was parried by Lee Grant.
There was a shock in the evening’s other quarter-final as misfiring Premier League side Southampton crashed out after losing 1-0 to third-tier Sheffield United at a raucous Bramall Lane.
After a promising start to the season, Southampton have lost their last four league games and their dismal run continued as Marc McNulty’s 63rd-minute goal sent Nigel Clough’s side into the last four.
McNulty bundled home after Southampton goalkeeper Fraser Forster fumbled Bob Harris’s fierce free-kick and the visitors finished with 10 men after Florin Gardos was sent off for hauling down McNulty.
It is the second season in succession that United have reached the last four of a major tournament, following their run to the FA Cup semi-finals last season.
Beleaguered Liverpool face a tricky assignment at Championship leaders Bournemouth on Wednesday, while Tottenham Hotspur host Newcastle United in the only all-Premier League tie of the round.