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Second-half goals from ?35 million striker Andy Carroll and Uruguayan international Luis Suarez sealed the points for Kenny Dalglish's men at Goodison after Everton midfielder Jack Rodwell was controversially sent off.
After a bright opening by Everton, the complexion of the match changed in the 23rd minute when Rodwell was given a straight red by referee Martin Atkinson for a robust challenge on Suarez.
The Uruguayan was left writhing on the floor in agony following the tackle but replays showed Rodwell clearly played the ball.
Rodwell protested his innocence to no avail before leaving the pitch, the 20th player to be dismissed in Premier League encounters between Liverpool and Everton, and the 11th in the last 14 games between the Merseyside rivals.
It got worse for Everton shortly before half-time when Phil Jagielka clattered into Suarez from behind for a certain penalty.
Dirk Kuyt's spot-kick was low and well-placed but Tim Howard guessed correctly to pull off a fine diving save as Everton breathed again.
Howard was well beaten soon afterwards when Charlie Adam crashed a 25-yard thunderbolt off the underside of the bar but again Everton escaped.
The deadlock was finally broken by Carroll in the 71st minute shortly after Dalglish had sent on substitutes Craig Bellamy and Steven Gerrard.
Bellamy surged forward and released fullback Jose Enrique down the left who crossed from the byline. Kuyt cleverly ducked out of the way of the cross for Carroll, who hooked his shot into the corner beyond Howard.
Liverpool wrapped up the points on 82 minutes when a dreadful defensive mix-up involving Sylvain Distin and Leighton Baines gifted Suarez a straightforward finish to make it 2-0.
In later Premier League fixtures Saturday, leaders Manchester United face Norwich at Old Trafford while Manchester City take on Blackburn.
Aston Villa face Wigan at Villa Park with Sunderland facing West Bromwich Albion at home and Wolves taking on Newcastle at Molineux.