Richarlison's third minute strike and Gylfi Sigurdsson's late penalty piled on more Anfield misery for Liverpool as they got their first win across Stanley Park since 1999.
Thiago was caught in possession and James Rodriguez threaded the Brazilian through who found the bottom corner.
Liverpool enjoyed a lot of pressure in the second half but it came to no avail as Chris Kavanagh awarded Everton a penalty as Dominic Calvert-Lewin was brought down by Trent Alexander-Arnold.
Sigurdsson made no mistake despite Alisson's best efforts.
It was the same old story for Liverpool as they lacked creativity and the injury list increased as captain Jordan Henderson limped off in the first half.
The Injury Crisis Continues as Everton Take Early Lead
Within three minutes, Liverpool were behind as Richarlison raced through on goal and put the Blues ahead.
Two good chances fell for the hosts as Roberto Firmino's effort was well blocked by Michael Keane and deflected wide.
Skipper Jordan Henderson volley was brilliantly tipped round the post by Jordan Pickford.
Liverpool injury woes continued as Henderson limped off with what looked like another muscle injury, inside half an hour.
20 minutes in a flurry of chances putting Everton under pressure, Pickford with two good saves.
Thiago careless in possession and the visitors capitalised.
The slow start continued from the Reds, despite going against them Everton with the wind in their sails.
Henderson injured, another muscle injury - when will this end?
Lacking creativity in the midfield, front three passengers in the first half.
First Half Reflection
Liverpool Pile on The Pressure But Sigurdsson Steals The Show
Liverpool came out with the intensity they were lacking in the first half.
The best two chances fell to Mane, both with the winger's head. Firstly he headed straight at Jordan Pickford before the Senegal international headed over moments later.
Jurgen Klopp's side had more possession and enjoyed a lot of the ball.
Mohamed Salah came close but was denied by the sprawling Pickford.
All the hardwork was undone as Dominic Calvert-Lewin was ruled to have been brought down by Trent Alexander-Arnold.
Sigurdsson sealed all three points with his penalty.
Liverpool finally showing intensity they lacked in the first half
Failed to find the back of the net.
Everton struggling to deal with the early pressure
Too predictable at times.
English referees again. No idea what Kavanagh saw there.
Second Half Reflection
LFC Transfer Room Verdict
Priyasha said: "This result is sickening. Top four looks far now. From the get go, the team lacked desire. We were good in spells but that did nothing for us, the loss of Jordan Henderson the final blow to an already miserable first half.
Trent was the only player who showed heart today, and was not at fault for the penalty at all, which should have been turned over, but, can't complain about VAR when we play like this.
"Our only chance of playing UCL next season seems to be only through winning the UCL, which is a bleak possibility. Very bleak."
Ritchie said: "It’s always hard when you lose to your rivals. It’s a lot harder when 3 centre backs and 2 midfielders playing as centre backs are all injured.
The Henderson injury was a real twist of the knife today. Everton managed the game well in the first half after going one up and whilst it was never a penalty, we didn’t do enough in the final third.
"On to the next one, where CL qualification is looking more and more difficult. We go again. YNWA."
Another deserved away win. Same old story I'm afraid. We lacked creativity again, been crying out for Henderson in midfield for weeks now looks like he will be sidelined for an extended period.
Leizpig midweek was all well and and good two goals but in the grand scheme of things, they never created and that was the case here.
Thiago careless in possession proved costly then from there we had bright moments but nothing clearcut.
We have to hope now that results go in our favour but first Liverpool need to score and win.
But this now looks like a regular occurrence between now and May as the struggle looks set to continue.
You can put this down to injuries, lack of fans but Liverpool look flat, lifeless and with no answers.