The Reds hit the back of the net four times as they showed their superiority over their fierce rivals.
But second half strikes from Salah and Diogo Jota meant Liverpool ran out 4-1 victors.
It was a tough day for Everton boss, Rafael Benitez.
Benitez is a legend at Liverpool, having guided them to four titles, including the Champions League, during a six-year spell from 2004-2010.
His history with Liverpool meant a large portion of Everton fans weren't exactly ecstatic when he was appointed their manager in the summer.
His cause hasn't been helped as Everton have struggled massively to start the campaign.
He had to stand there and watch his side get dismantled once again by their fierce rivals.
But Jurgen Klopp didn't rub it in as he appeared to be mindful for Benitez's struggles after Henderson's opener on Thursday.
The camera panned to the German manager to capture his celebration when Liverpool's captain gave them the lead.
And football fans think he muted his celebrations after taking a look art Benitez. Watch the moment below...
If Klopp did mute his celebrations because of Benitez - and it looks like that was the case - then that is a classy move.
Football fans have been reacting to his actions and you can view some of the comments below...
Benitez showed before the game that the respect between the two managers is mutual.
He wrote in his programme notes, per the Liverpool Echo: "Evidently, Jurgen has done a fantastic job and he is making the most of such a long time with his club.
“The ideas have developed over some seasons and now everything is clear for the players. That is one of the reasons why we expect such a tough match.”News Now - Sport News