Everton edged into the top four thanks to the in-form Richarlison and third consecutive win with a clean sheet.
Carlo Ancelotti's men struggled to deal with the physical presence of Mbaye Diagne early on, but the Senegalese saw a good shout for a penalty waived away.
After a flat first-half performance, Ancelotti could afford the luxury of bringing on midfielders Allan and Gylfi Sigurdsson in the second-half and the latter's quality delivery created the only goal.
Sigurdsson's pinpoint cross found Richarlison's head as the Brazilian netted for the fourth time in as many games after having scored just twice in the league this season before that run.
Diagne thought he had levelled in stoppage time, only to be denied by the offside flag as Sam Allardyce's men remain nine points adrift of safety.
Everton's stay in the top four could be short-lived if there is a winner between Liverpool and Chelsea later at Anfield.