Dominic Calvert-Lewin dashed West Ham's dreams of the Champions League as Everton battled to a 1-0 win at the London Stadium.
Calvert-Lewin's first-half strike was enough to secure three points for the Toffees and keep them in the hunt for Europe.
It was Everton's 11th away win of the season, with only their poor home form preventing them from being in the mix for the top four themselves.
Hammers manager David Moyes turned Everton from relegation candidates to the Champions League in 2005, but his chances of repeating the trick with West Ham are now fading fast.