Jurgen Klopp said pre-match that the reds would take this tournament seriously and they continue to do that as they easily moved into the next round.
Sadio Mane, Georginio Wijnaldum and Adam Lallana all had an opportunity to rest form the reds midfield and attack as Firmino, Lucas and Marko Grujic came into the side.
At the back Ragnar Klavan returned from injury but German keeper Loris Karius, now recovered from a broken bone in his hand, was named as the goalkeeper. A rare start too for Alberto Moreno who has been edged out by James Milner at left back.
Loris Karius kept a clean sheet on his Liverpool debut and the reds were good value for a deserved 3-0 win against Championship side Derby.
The reds dominated both halves but were leading by just the one goal at half time.
That goal arrived from a Coutinho corner - that Ragnar Klavan fired into the net after some poor defending. The Estonian captain could barely believe the ball had fell at his feet but he made sure with a left footed swing inside the area.
After the half time break the reds put the game to bed with two quick second half goals.
The impressive Firmino feeding Coutinho to fire in a second before Firmino and Coutinho combined again - this time freeing Divock Origi to send in a powerful shot for Liverpool's third.
Youngster Ovie Ejaria came on to make his LFC debut, Danny Ings made his first appearance from the bench for the season and Emre Can was also a second half subsitute replacing Jordan Henderson.
The Liverpool boss wasn't 100% happy with some aspects of the performance, suggesting the reds may have switched off in the final 10 minutes, but this was a fairly comprehensive cup win over a Championship side.
8 goals now in their two games away in this competition and the reds move into the fourth round which will be played towards the end of October.
Next up is a Saturday return to Anfield for only the second time this season - with Hull City aiming to stop Liverpool reaching a six game unbeaten run.