By the end, Chelsea fans were singing about Wembley again. Chelsea take the League Cup very seriously under Jose Mourinho and they moved into the semi-finals with only occasional discomfort here.
Eden Hazard, Filipe Luis and Andre Schurrle scored the goals, although Craig Bryson gave Derby brief hope when they trailed 0-2. There was plenty of spirit from Steve McClaren's side, and another promising cameo from Will Hughes, but class told here, the class of Hazard in particular. The Belgian was bewitching in possession at times, creating as well as scoring.
There was controversy too with Jake Buxton dismissed to Derby's bemusement in the second half. Earlier, Chelsea had been angered by the Derby stretcher-bearers' response to an injury to Kurt Zouma, the Chelsea centre-back, who suffered a dislodged tooth after being felled by Petr Cech as the goalkeeper cleared a corner.
Mourinho had every right to hurry up the stretcher-bearers as they ambled towards the pitch. Having endured that distressing Cech accident at Reading, Chelsea are particularly mindful of head injuries.
Zouma received good attention from the Chelsea medical staff but the lack of urgency on the behalf of the stretcher-bearers was disconcerting. No wonder Chelsea's players were urging them to speed up; Cesc Fabregas and Cesar Azpilicueta even grabbed the front of the stretcher to pull it quicker towards their stricken team-mate.
The other main point of the first half was Hazard giving Chelsea the lead. McClaren's side were given a reminder in the first half of the -dangers of dwelling in possession at the back when faced by hungry, well?drilled Premier League talent.
They were taught a painful lesson by Hazard, who exploited some lax play by Richard Keogh. It had been the Derby centre-half's poor clearance in the play-off final that proved so costly, gifting the ball to Queens Park Rangers' Bobby Zamora.
When Keogh decided to over-elaborate at the back, Didier Drogba pressed him, Hazard pinched the ball and Chelsea went for the jugular. Hazard rolled the ball back to Fabregas and darted into the box, collecting the return. Showing balance and a killer instinct, Hazard waited until the sliding Buxton committed himself before placing the ball effortlessly past Lee Grant.
As Hazard celebrated with his Chel-sea colleagues, the statisticians were revelling over Fabregas's involvement, revealing that this was his 15th assist in all competitions for Chelsea this term.
Derby had lost against Chelsea in the FA Cup last season, a defeat that McClaren had used to spur them on in the Championship, although they ultimately fell so agonisingly short in the play-off final with Keogh's error at Wembley. Breaking off from another tense promotion race, Derby had known that this quarter-final was another test of their potential. Chelsea were too good.
Mourinho had rotated to an extent but Chelsea were still strong, including five established first-teamers, John Terry, Azpilicueta, Fabregas, Nemanja Matic and Hazard. The -others were hardly below-par, -ranging from Cech in goal, Luis at left-back, Zouma at centre-half, John Obi Mikel anchoring with Matic, Schurrle on the right and Drogba up front.
Derby were outclassed. There was an old-school look to their defence in the numbers, two, six, five, three (Cyrus Christie, Keogh, Buxton, Craig Forsyth). They had lined up 4-1-4-1 with Omar Mascarell holding, Hughes probing from central midfield, hounding Chelsea players like Matic, and -building towards an attack led by Chris Martin. McClaren's side revealed few fears in the opening stages, and a scuttling break from Johnny Russell was stopped crudely by Schurrle.
The game was slightly scrappy, barring some elegant touches from Hughes, some good passes from Fabregas and wriggling runs from Hazard before the Belgian struck after 23 minutes. It was a reminder of the gap in class, a mistake by the Championship side clinically exploited by the Premier League leaders.
Derby actually rallied after -Hazard's goal. Hughes was trying to impose himself, bringing those subtle gifts to bear. Chelsea simply stepped up a gear again. Luis almost scored but shot straight at Grant.
Then came that Zouma incident. Cech came flying out to meet Russell's corner, punching the ball clear but catching Zouma with his shoulder, dislodging a tooth. After a lengthy stoppage, Branislav Ivanovic came on.
Chelsea remained in control, barring a scare when Russell crossed from the right and Hughes met the ball first time but Cech was well-placed to block. Chelsea stepped up again. Gifted a free-kick opportunity 30 yards out, to the right, ideally positioned for a left-footed player. Luis ran in and swept the ball past the wall and past Grant for an exquisite goal, a spectacular way to open his Chelsea account.
Chelsea then lost the hobbling Drogba, who was replaced by Loic Remy. Derby refused to give up. Hughes tried to open up the Chelsea defence and then, to the utter delight of the Derby fans, Bryson found a way through after 71 minutes. Russell cut the ball back, and Bryson calmly finished from 20 yards, shooting from right to left past Cech.
Derby briefly believed. Their fans backed them unconditionally, willing them to get level. Within seven minutes, Derby suffered a huge setback, checking their momentum.
Hazard sent Remy flying through the middle, running shoulder-to-shoulder with Buxton, who put an arm across, clearly impeding the Chelsea striker.
As the ball ran clear, Grant slid out, seeming to handle just outside the area. Mourinho brandished an imaginary card. A real one soon appeared.
The assistant referee signalled an offence and Jon Moss ran across, returning and running towards Buxton. He sent the defender off, enraging Derby.
It got worse for Derby. With eight minutes remaining, Fabregas picked out Remy, whose shot was saved by Grant. Schurrle was quickest to the loose ball, slamming it into the net and confirming Chelsea's passage into the semi-finals.
When Moss the blew the final whistle, Chelsea fans began singing again about Wembley. The semi-finals await first though.