The contest was played in an electric atmosphere at Wanda Metropolitano Stadium here on Thursday night where almost 65,000 fans responded to the call by suspended Atletico coach Diego Simeone the day before to serve as a true 12th man and will the home side to victory, reports EFE news agency.
The former Chelsea striker's goal proved to be the victor, following the 1-1 draw at the Emirates a week ago.
Defeat for the Gunners means they will miss out on Champions League football for the second season in succession, and, somewhat more agonisingly, that Wenger's tenure in charge ends with him never quite being able to lead the club that he loves to European glory.
The gunners managed just one shot on target to rescue a season that was not enough.
But Costa, so often a thorn in the side of Arsenal during his time at Chelsea, tucked away a chance on the stroke of half-time to seal a 1-0 win and safe passage to the final.
Asked about Koscielny's chances of recovering in time for the World Cup, Wenger said: "That depends on the scan". He is a World Cup victor and these are crocodile tears I am seeing from him and he is not conning me."Wenger said he "does not agree" with Keown's comments about Ozil's display and insisted the midfielder "gave a lot".He added: "We lose as a team. I'm very, very sad to leave the club with this exit".
The Frenchman offered a vehement defence of Mesut Ozil after former Arsenal defender Martin Keown lambasted the German's performance while working for BT Sport.
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"It will be a game where we know exactly what to do", Wenger said.
Arsenal manager Arsene Wenger said: "It doesn't look very good".
"He wasn't fit to wear the shirt for me tonight and I have seen this a lot this season and it needs to be said".
Wenger's 250th European fixture with Arsenal also proved to be his last, decided by a typically ruthless finish from Costa, who delivered the sort of barnstorming display defenders must have nightmares about.
Needing to score following a 1-1 draw in the first leg, Arsenal's task was made more hard when Laurent Koscielny was forced to leave the pitch with an Achilles tendon injury, which could put the Frenchman's place in the World Cup finals in danger.
"We are delighted to be in the final, we've wanted to be back in a final ever since the last one", Simeone added. "Along with the other forwards we have, that was very important".