Ragnar Klavan, Philippe Coutinho and Divock Origi were all on the scoresheet as the Reds took their tally to 19 in seven matches this season.
This was the fourth time they have scored three or more and Klavan was their 10th different scorer, highlighting the work Klopp has done in terms of strengthening the team not just in manpower but in the options available.
In two cup games this season they have racked up eight goals, albeit against SkyBet Championship sides.
The return of Roberto Firmino, who missed the win at Chelsea with a minor groin problem, highlighted how much he oils the wheels in the final third as his link-up play, particularly with Brazil team-mate Coutinho, and ability to run into the spaces was again top drawer.
Goalkeeper Loris Karius made his debut as one of seven changes from Friday’s win at Chelsea but, even with no Daniel Sturridge in the squad, it was by no means a weakened side.
Klopp insists he does not have a number one goalkeeper but Karius, a £4.7million summer signing from Mainz, wears that digit on his shirt and this was essentially the start of his bid to depose Simon Mignolet.
In truth the manager would not have learned much as, apart from late on in the first-half when the German reacted quickly to dash out of his area to dispossess Darren Bent, Karius was untroubled.
He did show an aptitude and composure with the ball at his feet, something Mignolet does not, but it was in the other penalty area where that was displayed best midway through the half.
Derby captain Richard Keogh made a hash of trying to clear Coutinho’s corner and Klavan reacted quickest to stab home from close range.
Liverpool’s half-time lead should have been greater as Derby, 20th in the Championship and struggling for goals, stood off their opponents and conceded large amounts of territory.
Goalkeeper Jonathan Mitchell, with good saves from Origi from close range and Coutinho from distance, kept it competitive until five minutes after the break.
The fit-again Firmino played an exquisitely- weighted return pass to his Brazilian team-mate whose low finish effectively ended the game as a contest.
All doubt was extinguished five minutes later when Origi smashed home an angled drive from Coutinho’s through-ball after Firmino won the ball back on the edge of Derby’s penalty area.
Mitchell continued to do his best, denying Firmino from close range, but with the result in the bag Klopp sent on Danny Ings for his first appearance of the season and handed a debut to promising academy 18-year-old Ovie Ejaria to further enhance his options.
Mitchell; Christie, Keogh, Pearce, Olsson (Lowe 64’); Baird (Ince 46’), Johnson, Butterfield, Hughes; Weimann (Elsnik 56’), Bent
Karius; Clyne, Matip, Klavan, Moreno; Lucas, Grujic, Henderson (Can 56’); Coutinho (Ings 63’), Firmino (Ejaria 77’), Origi