Mohamed Salah scored in the competition for the first time since Boxing Day and January signing Cody Gakpo notched his first goal for the club in the Anfield triumph.
Liverpools success ended a run of four league games without a win and lifted them to ninth in the table while struggling Everton remained in the bottom three.
Asked how he felt after the game, Liverpool manager Klopp said: Relieved thats it, thats the main feeling since the final whistle.
Im happy with the performance. It was necessary that we played the game we wanted to play and not the game Everton wanted to play.
You cannot avoid it all the time but they didnt have an extremely high number of set-pieces. With the all aggression you have to show, you have to make sure you dont overdo it because each set-piece is a massive threat and something they want to have.
I thought we did that really well. On top of that we were super-dominant, switched the sides, kept the ball and kept them running.
Both goals came from counter-attacks that is allowed as well and I loved both goals.
For both goals we had three options in the box. A lot of things were different tonight so it was our best game for a while.
We needed this game, we needed this performance. Our people absolutely deserved it. I loved the atmosphere even before we scored. It is an insane atmosphere people create here.
Everton manager Sean Dyche felt his side started strongly but did not recover after being caught out for Salahs 36th-minute opener, which came seconds after James Tarkowski hit the post at the other end.
Dyche said: I think they had the upper hand in the early stages without opening too much up.
Our shape was good but then it changed in 15 or 16 seconds. We hit the post and didnt react as quick.
We know Liverpool are good in the transition phase and we conceded a goal without needing to.
At half-time we said we had to use the ball better but then if you give away another goal so quickly it is very difficult, especially here.
We kept plugging away but there were too many sloppy passes and too many turnovers.
Dyche insisted that, as he had not got too excited after victory over Arsenal in his first game in charge, he was not downbeat about derby defeat in his second.
He said: The players have taken a lot on board so I am not going to be too critical. They have been working very hard, very diligently.
I know the importance of this game to Evertonians but, on paper, youd take three points from these first two games. Thats a start point.
I dont remember being too high after we beat the leaders. I was quite balanced about it and Im quite balanced tonight.