Carlos Alcaraz’s header earned the Toffees victory on Tyneside.
However, the Magpies finished fifth after Aston Villa were controversially beaten 2-0 at Manchester United.
Villa’s hopes were dented when goalkeeper Emi Martinez was sent off just before half-time for bringing down Rasmus Hojlund outside his box.
Yet the visitors thought they had taken the lead at Old Trafford, only for Morgan Rogers’ effort to be ruled out for kicking the ball out of the hands of Altay Bayindir.
Villa were furious at VAR’s decision not to reverse the call and were hit with a sucker punch moments later when Amad Diallo headed in Bruno Fernandes’ cross.
Christian Eriksen’s penalty sealed the win as United finished the campaign in 15th.
A place in the Europa League is Villa’s consolation for finishing sixth.
