The Gunners lost 1-0 in the Wanda Metropolitano stadium courtesy of a goal from ex-Chelsea striker Diego Costa in first-half stoppage time, a week after they were held to a 1-1 draw in north London.
Arsenal missed a whole host of chances to put themselves in a commanding position in the tie in the first leg, with Diego Simeone’s side a man down after just 10 minutes.
Arsenal went into the second leg in Madrid knowing they had to score due to the away goals rule, but an attack spearheaded by Alexandre Lacazette managed just seven shots.
Simeone’s men have not lost at home in the league all season and not conceded a goal since January, and Arsenal failed to threaten goalkeeper Jan Oblak as much as they did back home.
At the other end, the Gunners’ defence was characteristically nervy from the off and captain Laurent Koscielny was taken off with what looked to be a serious Achilles injury after just 12 minutes.
Calum Chambers replaced him and performed well, with his more experienced teammates culpable on Costa’s goal when it arrived in the second minute of time added on before the break.
Granit Xhaka’s weak header put Arsenal on the back foot and Antoine Griezmann – who equalised late at the Emirates – threaded through the Spaniard, who held off Hector Bellerin with ease before finishing past David Ospina.
The 2-1 aggregate scoreline means there will be no fairytale finish for long-serving boss Wenger, who was hoping bring the curtain down on his 22-year tenure at the Emirates by leading his side out in the final in Lyon on 16 May.
His final game in the Arsenal dugout will instead be a rather less glamorous trip to Huddersfield on the final day of the Premier League season three days earlier, with his last home match coming up this Sunday against Burnley.
Looking close to tears in his post-game press conference, Wenger, 68, admitted: “I’m very, very sad to leave the club with that exit. It will take some time to recover from that.”
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Failure to win the Europa League also means Arsenal will endure a second consecutive season without Champions League football, with the club already confirmed to have missed out on a top-four finish.
The final will be contested by Atletico and Marseille after the French side snatched a dramatic extra-time goal in the second leg of their tie against RB Salzburg to win 3-2 on aggregate.