At the other end of the table, Brighton and Watford did little to boost their chances of beating the drop in a 1-1 draw at the Amex Stadium.
Goals from Bernard, Richarlison and Dominic Calvert-Lewin at Goodison lifted Everton up to seventh and within five points of the top four.
Everton have now won five and drawn two of eight league games since Ancelotti took charge in December.
"Since I arrived the team did a fantastic job," Ancelotti told BT Sport. "Now we have a possibility to think about Europa League".
By contrast, Palace are looking over their shoulders at the relegation zone.