Despite his 10-man team beating Everton 2-1, rivals Liverpool and Manchester City won their games to stay ahead of them in fifth.
It is the first time the Arsenal finished outside the top four of the Premier League in 21 seasons under Arsene Wenger since 1995-96.
The Frenchman, who leads his team into Saturday’s FA Cup final against Chelsea on Saturday, said after the game: “It’s very sad because for 20 years we were in it. For 75 points we are one point short.”
Liverpool, meanwhile, will return to Champions League for the first time since 2014-15.
And coach Jurgen Klopp couldn’t help but rub it in after his team beat Middlesbrough 3-0. The German said: “It feels good in this moment. Everything feels good in this moment at this club.
“We had an outstanding season, 76 points, lots of injuries but we qualify for Champions League ahead of Arsenal and Man United.”
Pep Guardiola’s City sealed third place in the league with a 5-0 win at Watford and warns that it will just get more difficult to make the top four in the future.
He says: “I’m happy to be third. It is going to get tougher and tougher to get into the Champions League.
“Of course we want to be champions. Qualifying for the Champions League is important for the club and we will come back stronger.”