Arsenal could not protect a 2-1 lead against 10 men as Joao Palhinha's 87th-minute strike rescued a point for Fulham at the Emirates.
Everton boss Sean Dyche has dismissed talk of a crisis at the club after defeats in their opening two matches left them bottom of the table.
Everton paid the price for a string of missed chances as their goalscoring issues were exploited by Fulham in a 1-0 defeat at Goodison Park.
There will be a London derby between Fulham and Tottenham in the second round of the Carabao Cup.
Willian has re-signed for Fulham on a one-year deal with an option for a further 12 months.
Manchester United sealed third spot as Bruno Fernandes' smart finish completed a 2-1 comeback win against Fulham on the final day of the Premier League season.