The other scorers in the rout were Salomon Kalou and Joe Cole, with David Jones hitting the solitary reply for the relegation-haunted visitors.
The demolition almost started with Chelsea's first attack on three minutes when fine set-up play by Nicolas Anelka and Kalou fed Lampard, but his shot ricocheted off the goalkeeper's upright and was hastily cleared.
Derby, who are rooted to the bottom of the Premier League, were happy for the first quarter of the match to sit deep behind the ball and allow Chelsea total possession.
Derby were forced into an early substitution with Alan Stubbs coming off for Dean Leacock on 16 minutes.
But on 28 minutes, Leacock upended Lampard inside the penalty area.
Up stepped the England midfielder, who slotted home his 104th goal for the Blues by sending Carroll the wrong was and nestling the ball into the left-hand corner of his net.
Kenny Miller had a looping header for the visitors on target but well saved by Carlo Cudicini two minutes later.
Goal number two came four minutes from half-time and it was self-inflicted.
Carroll came racing out of goal to intercept a through ball but instead slid the ball into the gleeful path of Kalou 25 yards out and he had the simplest of chances to send the ball back into the empty net
Chelsea continued in the same vain in the second half by dominating possession and leaving Derby to hack the ball clear for some respite.
Derby boss Paul Jewell replaced midfielder Hossam Ghaly with striker Robert Earnshaw on 52 minutes.
But before Jewell could bring on Jones for Eddie Lewis, Chelsea hit goal number three on 57 minutes.
Joe Cole zipped through and cut back to Lampard three yards out and he tucked the ball home comfortably.
Joe Cole scored the fourth goal on 65 minutes. Claude Makelele sent Anelka clear and the Frenchman forced a low left-handed save from Carroll, but Cole reacted quickest and slotted home from ten yards.
Sixty seconds later and Lampard completed his hat-trick after belting a right-footer from the edge of the penalty area which bounced over the hapless Carroll and into the net.
Avram Grant instantly brought on Didier Drogba for Michael Ballack
Lampard hit his fourth of the night and Chelsea's sixth with a low left-footed rasper on 72 minutes from the edge of the penalty area.
Remarkably, Derby broke quickly a minute later and Jones found himself free eight yards out and slid a neat left-footer past Cudicini for an undeserved consolation goal.