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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.
Manchester United's top-four hopes were dented as they were held to a 0-0 draw by relegation-threatened Watford.
He takes over a club four points above the relegation zone, and facing the distinct possibility of finding themselves competing outside top tier English football for the first time in almost 70 years,
The result lifted Palace above Everton in the Premier League standings as they moved up two places to 12th.
Chelsea lead the table on 29 points, with City three points behind and Liverpool one further back in third.