Arsenal will once again rue missed chances from the first-leg where they were a man up for most of the tie, while Costa’s ruthless finish and overall buccaneering display highlight what then Gunners have been missing in their ranks.
Wenger’s final season in north London will end with his team missing out on Champions League qualification for a second year in a row, a devastating low point as the club heralds a new era.
A tough task for Arsenal got worse when Laurent Koscielny, clutching his left Achilles, was stretchered off in the 12th minute.
Costa, in predator mode, had bullied the Arsenal backline all night, knocking over Nacho Monreal before firing wide, after bullying Koscielny in a previous tussle.
The deadlock was broken just before half-time after Granit Xhaka’s weak header allowed Antoine Griezmann to thread a pass through to Costa.
The Spain international he held off Hector Bellerin with ease before coolly finishing past the shaky David Ospina.
Aaron Ramsey came close for the Gunners but the ball would not fall for him six yards out while Mesut Ozil flashed too sumptuous deliveries across the face of goal.
Griezmann missed a glorious opportunity to end the tie after Costa had turned Shkodran Mustafi inside out but the defender recovered just in time to block.
Arsenal surged forward late on in search of an equaliser to send the tie into extra-time, but Atleti haven’t conceded at the Wanda Metropolitano since January and they easily closed out the tie to seal their third European final in four years.
Diego Simeone’s men will be taking in former European champions Olympique Marseilles in the final after the French side edged Austria’s RB Salzburg 3-2 aggregate despite losing 2-1 on the night.