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The champions were humiliated 7-2 by Aston Villa just before the international break but they quickly set about putting things right when Sadio Mane scored in the third minute, firing home a low cross from Andy Robertson.
Liverpool suffered a blow, though, eight minutes later when defender Virgil van Dijk went off injured after a reckless challenge from Everton keeper Jordan Pickford.
Everton took advantage of the Dutchman's absence, when, from a corner, Michael Keane rose at the near post to power a header past Adrian.
Liverpool were on top for long stretches and Mohamed Salah restored their lead in the 72nd minute, pouncing on a headed clearance from Yerry Mina and drilling a drive past Pickford.
But Everton, determined not to lose their unbeaten record, got back on level terms when James Rodriguez sent Lucas Digne down the left and his perfectly floated cross was headed home by the leaping Dominic Calvert-Lewin.
Everton's Brazilian forward Richarlison was sent off in the 90th minute after a wild challenge on Thiago Alcantara.
Liverpool thought they had won the game in stoppage time through Henderson but VAR ruled that Mane had been narrowly offside and the goal was chalked off.
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