Bernard, Dominic Calvert-Lewin and Richarlison on target as Everton edge Crystal Palace at Goodison Park
The Brazilian finished decisively after 58 minutes before Jordan Pickford redeemed himself for an earlier blunder with a point-blank save to deny Christian Benteke a second equaliser.
Bernard had earlier given Everton the lead before Benteke levelled with his first goal since April.
Daniel Calvert-Lewin wrapped up the points for his side with a late third goal for the Toffees.
Everton are now seventh on the log with 36 points while Crystal Palace remained 14th with 30 points.