The defender's superb free kick and Eden Hazard's opener made it a comfortable night for Jose Mourinho at the iPro Stadium.
Derby briefly rallied after Craig Bryson pulled a late goal back but their hopes ended when Jake Buxton was sent off for a foul on Loic Remy and Andre Schurrle added a third.
The visitors lost Kurt Zouma after a serious-looking collision with goalkeeper Petr Cech but they were in cruise control and rarely got out of third gear.
It was a reminder to the Rams, should they earn promotion to the Barclays Premier League, of the challenges they face.
Jose Mourinho issued his intent as Cesc Fabregas, Cesar Azpilicueta, John Terry and Didier Drogba - who came off injured - started while Derby boss Steve McClaren made just two changes.
Buxton came in for the suspended Ryan Shotton in defence while Will Hughes replaced Jeff Hendrick in midfield.
Third in the Sky Bet Championship, the Rams competed well with the top flight leaders without creating any openings while Chelsea were comfortable despite failing to threaten themselves.
But County hit the self destruct button on 23 minutes to gift the visitors the lead.
Richard Keogh was robbed on the edge of the Derby box by Hazard, he played a one-two with Fabregas and the Belgium international danced past three challenges to fire into the bottom corner.
It was ruthless Chelsea's first shot on target
The goal settled them as Luis immediately fired straight at Lee Grant.
Aside from a flurry of corners late in the half Cech was a spectator, until he clattered into Zouma while he punched away the final corner a minute before the break
The defender was poleaxed and needed treatment but while their club doctors attended to the defender Chelsea were frustrated by the stretcher-bearers.
Angered by their slow speed Fabregas and Azpilicueta grabbed the stretcher and helped carry it to their stricken team-mate.
Branislav Ivanovic replaced him as half-time came, which gave Derby chance to regroup.
And they came close soon after the break when Hughes met Johnny Russell's cutback but Cech pushed his low effort wide.
The chance briefly reinvigorated the Rams only for Chelsea to brilliantly extend their lead on 56 minutes.
It seemed too much to ask when Luis lined up a 30-yard free kick after Drogba was clipped, only for the defender to fire in a wonderful effort which flashed past Grant.
Again, Derby were silenced as Chelsea controlled the pace and tempo but the Rams grabbed a lifeline out of the blue with 19 minutes left.
Russell found Bryson on the edge of the area and the midfielder curled in off the post to give the hosts hope.
It disappeared though, on 78 minutes when Buxton was dismissed after he pulled Remy back as the striker ran through, much to the shock of his team-mates.
And, four minutes later, Chelsea wrapped up the win when Grant saved from Remy and Schurrle fired in the rebound.
Source : PA
Source: PA