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Liverpool's advantage was doubled by Hugo Ekitike with a calm effort before the break.The Reds were forced to hold in the second half after Idrissa Gueye pulled one back for the Blues.
At least it shows they are capable of winning in multiple ways, having been forced to net late winners in their prior five matches in all competitions.
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Here, the ECHO looks at the ongoing reaction to Liverpool's win in the latest chapter of Merseyside derby history.
Already this season the Reds have allowed two-goal leads to slip against Bournemouth, Newcastle United, and Atletico Madrid.
If they go 2-0 up or they start winning the game, they're comfortable, they just seem to take their foot off the gas - they're not as vicious and ferocious as they are in that first 20 minutes or half an hour," the former winger told talkSPORT.
"They get a little sloppy and comfortable, that's when teams smell a little bit of blood and get themselves back into the game.
"You felt at any stage Liverpool can take it up a couple of notches."
That said, Pennant carried on and believes the reigning champions are bound to fix their glaring issue in the near future.
"They haven't [got going yet].
"He's played in patches.
"His quality is there to be seen when he's having his good little spells.
I think in midweek (against Atletico Madrid) they were really good bar the little spell in the second-half.
"Today (against Everton) the more obvious thing was the fatigue, but still so many new players forming relationships and understanding what each of them does."
Claiming that Slot's new forwards are still adjusting too, Murphy continued: "There are still so many new players forming relationships and understanding what each other does.
Ekitike is different to [Alexander] Isak - so if you're playing with either one you have to learn their traits.
"So to win five our of five without playing at your best is a phenomenal start for them and actually is a really good sign."
Although Ekitike landed the telling blow on his beloved Everton, former striker Wayne Rooney was left impressed.
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"Ekitike, he's a good player.