Striker Dirk Kuyt's 50th Liverpool goal gave his side a 1-0 victory in a fiery 213th Merseyside derby at Anfield which saw both teams reduced to 10 men.
The Dutchman scored for the fourth time in as many games with a close-range header 10 minutes after half-time.
Liverpool played the last 55 minutes at a numerical disadvantage after Sotirios Kyrgiakos was dismissed for a two-footed challenge on Marouane Fellaini while Steven Pienaar was shown a second yellow in second-half injury time.
The red cards took the tally to 19 in the last 13 years but referee Martin Atkinson somehow missed Marouane Fellaini's stamp on the Greek defender's right shin in the same incident which led to his red card.
It threatened to overshadow a fiercely-contested encounter until Kuyt struck early in the second half to ensure Everton's winless run at Anfield since 1999 continued.
Liverpool have now gone seven Barclays Premier League matches without defeat, keeping six clean sheets in the process. By contrast, Everton's nine-match unbeaten run came to an end and they have now won just two of the last 21 league derbies.