Elsewhere, there were wins for Arsenal, Man United and Aston Villa.
Everton have guaranteed their survival in the Premier League.
A second half goal by Abdoulaye Doucoure gave them a 1-0 victory over Bournemouth at Goodison Park.
And that was enough to relegate both Leicester City and Leeds United.
Leicester City defeated West Ham 2-1 at the King Power Stadium, but they go down to the Championship seven years after winning the Premier League.
Leeds United were beaten 4-1 by Tottenham and return to the second tier after three years in the top flight.
Elsewhere, Brentford beat champions Manchester City 1-0
It ended Arsenal 5 Wolves 0, Aston Villa 2 Brighton 1, Chelsea 1 Newcastle United 1, Crystal Palace 1 Nottingham Forest 1, Manchester United 2 Fulham 1.
And relegated Southampton 4 Liverpool 4.
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