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Everton's first derby victory at Goodison Park in more than 13 years prevented Liverpool from climbing level with Arsenal at the top of the league table. Juergen Klopp's men are three points behind the Gunners (77) with four games to play.
The heavy defeat leaves Everton 16th on 27 points, two places and two points above the relegation zone.
Manchester United fanned the embers of their top-four hopes with a 2-0 win against visiting Everton.
Pep Guardiola's newly-crowned FIFA World Club champions had lost ground in the title race after a run of one win in six league games.
Everton will extend their unbroken 69-year stay in the English top flight as midfielder Abdoulaye Doucoure's stunning strike earned them a 1-0 victory on the final day of the season.
James Tarkowski's headed goal made the difference in the premier league encounter on Saturday.
Lampard's last game in charge was Saturday's 2-0 league defeat at West Ham United, which kept Everton second-bottom of the table on 15 points after 20 games, above Southampton on goal difference, and two points adrift of the safety zone.
The match ended goalless with both sides hitting the woodwork.
Vieira was confronted by a fan as he was walking towards the away dressing room after the final whistle. The fan appeared to provoke Vieira, who then kicked him to the ground and had to be led away by other supporters.
Chelsea kept a grip on third place but needed a last-minute goal from American Christian Pulisic to beat West Ham United 1-0 at Stamford Bridge.
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