The Brazilian attacker scored one and set up another as the Reds added to the misery of the home side and marched into the fourth round of the competition.
Philippe Coutinho scored one goal and assisted the other two
as Jurgen Klopp's much-changed side crushed their Championship
opponents, in the process extending their unbeaten run in all
competitions to five matches.
Ragnar Klavan grabbed Liverpool's opener, the
defender capitalising on a defensive error by Derby's Richard Keogh to
find the net in the 24th minute.
Liverpool then hit their hosts with a quick-fire double at the start
of the second half to put the contest to bed, Coutinho slotting home a
one-on-one opportunity before setting up Divock Origi to do likewise
from a tight angle.
Having brushed aside Burton Albion in the last round, Klopp's side
have now managed eight goals without reply in the competition, raising
hopes last year's beaten finalists could be set for a return to Wembley.
The German shuffled his pack for the trip to the east
Midlands, handing full debuts to goalkeeper Loris Karius and midfielder
Yet despite making seven alterations to the XI that started the 2-1
win over Chelsea last time out, Liverpool’s new-look line-up were
quickly into their stride.
Jordan Henderson and Roberto Firmino both fired long-range shots just
over before an unlikely source rewarded the visitors' dominance with an
Keogh's fluffed clearance from a Coutinho corner allowed Klavan to
turn the ball in at the back post, with his left-footed effort finding
the net via a deflection off Cyrus Christie.
Derby goalkeeper Jonathan Mitchell tipped over a Coutinho free-kick
to keep the deficit down to one at the break, although he was unable to
deny the Brazilian from scoring five minutes after the interval.
Coutinho played a one-two with compatriot Firmino, whose wonderfully
weighted return ball allowed his team-mate to slot beyond Mitchell
without breaking stride.
The tie was put beyond Derby four minutes later, Coutinho this time
the provider as his ball down the right channel allowed Origi to hammer
the ball into the roof of the net.
With such a healthy lead, Klopp had the luxury of withdrawing captain
Henderson and star man Coutinho ahead of Saturday's league game against
Hull City at Anfield.
Danny Ings was given the chance to make his first appearance of the
season off the bench, while there was a debut for another substitute,
youngster Ovie Ejaria.
Karius did get a reprieve when his failure to deal with a corner saw
Keogh head just over, although it was a rare moment of concern on a
night that saw Liverpool in cruise control.