Two goals from Everton were enough to hand them a win, the first win at Anfield in 22-years as top four hopes for the Reds went down in ruins.
Richarlison was on target in the third minute when he utilized James Rodriguez pass and hand the visitors the lead.
Mohamed Salah, Sadio Mane and Jordan Henderson pushed to give Jurgen Klopp something to smile about but they were both denied by Jordan Pickford.
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England striker Dominic Calvert-Lewin was brought down in the 83rd minute as Everton were handed a penalty which was converted by Sigurdsson.
The defeat see Liverpool occupy sixth position with 40 points from 25 games and level on pints with Everton who are currently in seventh position.
The injury stricken Liverpool is set to suffer another blow after Jordan Henderson left the pitch limping.
Fellow Premier League giants Chelsea are now three points clear of Liverpool even after recording a poor string of results before replacing Frank Lampard with Thomas Tuchel.
Klopp will be forced to look for a quick solution or else suffer the biggest embarrassment from lifting the Champions League to failing to qualify come next season.