Ronald Koeman's side suffered a 2-1 defeat to Lyon in Goodison Park on Thursday, leaving them with just one point from their first three games.
Ashley Williams gave Everton hope when he cancelled out Nabil Fekir's early penalty, but Bertrand Traore consigned them to defeat with a goal 15 minutes before the end.
According to Opta, No English side has had such a poor record at this point in the tournament and it leaves them six points behind leaders Atalanta.
It puts the Toffees in a precarious position as qualification for the second round is now a huge challenge.
Only three sides have progressed to the next round after collecting a point in the first three games.
Although Koeman's men started the season with four consecutive clean sheets in the Europa League qualifiers and opening Premier League game, they have conceded in all but one of their last 12 games – a League Cup third round win over Sunderland.
The defeat to Lyon means Everton have gone three home games without a win for the first time under their Dutch coach, whose future will be placed in even further doubt.
The 54-year-old is reportedly close to being sacked and his team are now four games without a win in all competitions. They sit just two points above the Premier League relegation zone ahead of Sunday's game against Arsenal.