Everton, under new boss Roberto Martinez, won 4-1 at League Two side Accrington, who are managed by former Toffees striker James Beattie.
Darron Gibson gave Everton an 11th-minute lead, slotting in from a corner.
Victor Anichebe headed home a second from Kevin Mirallas' cross five minutes later.
Mirallas got on the scoresheet himself with a fine free-kick a minute before the break, but the hosts pulled one back on the stroke of half-time through Peter Murphy.
Leon Osman headed home a fourth for Everton shortly after the hour mark as they ran out comfortable winners.