Jurgen Klopp's side produced a dominant performance against the Championship side and the result never really looked in doubt.
The Reds took the lead halfway through the first half when Klavan slotted home from a Coutinho corner.
Coutinho then got himself on the scoresheet just after the break when he scored after a nice one-two with Firmino.
Divock Origi then put the icing on the cake when he smashed home a third goal for Liverpool just four minutes after Coutinho's strike.
It was a disappointing evening for Derby, who managed just two shots on target all game, adding to a poor start in the Sky Bet Championship.