Manchester City won 3-2 at Blackpool to move second in the table after Liverpool owners John W Henry and Tom Werner watched goals by Tim Cahill and Mikel Arteta condemn the five-times European champions to their fourth defeat of the season.
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Fernando Torres went closest for the visitors at Goodison Park when his first-half header was turned over the bar by Everton keeper Tim Howard, but Liverpool offered little else.
They slipped one place in the standings on goal difference, level on six points with Wolverhampton Wanderers and bottom-placed West Ham United.
Chelsea stayed top after a 0-0 draw at Aston Villa on Saturday. City are second, two points behind, Arsenal third after a 2-1 win at home to Birmingham City and Manchester United fourth after a 2-2 draw with West Bromwich Albion at Old Trafford.
HODGSON UNDER PRESSURE
Henry's New England Sports Ventures, owners of the Boston Red Sox baseball team, completed a takeover of Liverpool on Friday after contentious legal battles with the previous U.S. owners, Tom Hicks and George Gillett.
The jubilant scenes in London on Friday concluded one of the most dramatic weeks in Liverpool's 118-year history, yet the euphoria of the takeover battle did not carry over to the pitch.
Everton fans taunted their counterparts with chants of "Going down" after the defeat that left Liverpool 19th in the table, their lowest ever position in the Premier League.
Manager Roy Hodgson was given a vote of confidence by Henry before Sunday's match, but the defeat is certain to focus attention on the future of the Englishman who has presided over Liverpool's worst start to a season for over 50 years.
"I don't feel it to be a crisis. The way we played today - I don't think anyone would believe that's the level of football a team in the bottom three or four would play," Hodgson told Sky Sports.
"On the other hand, there's six points from eight games and that's a very, very poor return. We do need to start winning and climbing up that table soon, and until we do so I daresay the word 'crisis' will be bandied around."
DETERMINED RUN
The breakthrough came from Seamus Coleman's determined run into the area. The Irish defender's cross took a deflection and the ball sat up for Cahill to drill his finish past Jose Reina.
Everton doubled their lead five minutes after half-time, when Sotiros Kyrgiakos's headed clearance from a corner fell to an unmarked Arteta, whose right-foot shot from the edge of the area swerved viciously and left Reina clutching at air.
At Bloomfield Road, Carlos Tevez put City ahead after 67 minutes with a neat touch at the near post from James Milner's cross.
Marlon Harewood got the faintest of touches to a free-kick to bring Blackpool level with a header in the 78th minute, but City were back in front within a minute.
Argentine Tevez robbed defender Ian Evatt - who claimed he was fouled - and his shot took a deflection before finding the bottom corner.
Spain forward David Silva made it three with a splendid solo goal, selling two dummies in the area and curling home a left-foot shot. Gary Taylor-Fletcher's stoppag
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