Nicolas Pepe scored twice in the second half as Arsenal beat Brighton & Hove Albion 2-0 in their final Premier League game of the season on Sunday but Mikel Arteta's side failed to qualify for the Europa Conference League after finishing eighth.
On a frantic final day, Arsenal finished a point below north London rivals Tottenham Hotspur, who beat Leicester City 4-2 to seal seventh place and qualify for the new third-tier European competition.
Arsenal ended the season strongly with five wins on the trot to finish with 61 points, five more than last season, but they will not be involved in a European competition for the first time in 25 years.
"It's frustrating because last season with 61 points you were fifth and now you are eighth," Arteta told reporters.
Arsenal dominated possession in the first half and went close to opening the scoring but Rob Holding's goal from a set piece was ruled out for offside while his centre back partner Gabriel saw his floating header come off the crossbar.
Pepe gave Arsenal the lead four minutes after the restart, controlling a low cross from Calum Chambers with his left foot before firing a fierce strike with his right foot past Robert Sanchez in Brighton's goal.
Arsenal doubled the advantage when Martin Odegaard released Pepe down the right channel and the Ivorian forward composed himself to curl a low finish inside the far post for his 10th league goal of the season.
Odegaard returns to his parent club Real Madrid and Arteta said he hoped they had done enough for the 22-year-old to consider a return after an impressive few months at the club.
"We have a very clear opinion on what we would like to, but he's not our player," Arteta said.
"We tried to make everything that we could to get Martin performing for the team. Hopefully we have given him the feeling that it could be a good place for him."