Arsenal fashioned a number of openings but the final pass was too often lacking as they were unable to breach Atletico's defence.
The result also means Arsenal have lost hope of playing in the Champions League next season and will be forced to make do with another campaign of Europa League football.
Some said it might be time to sacrifice him.
'Even very frustrated, as well, because when you go out of a competition and you had the performances in 180 minutes that we had, it very hard to take'. I'm very sad tonight.
The Gunners could not respond, letting the Spanish giants claim a 2-1 aggregate win to book a spot in the Europa League final against French side Marseille.
Just 12 minutes later, Salzburg found another goal to increase their lead to 2-0 and equalized the tie at 2-2 on aggregate. Atletico played with fear in the first half that if we scored they could be out, but once they scored they played with a cushion.
"We're very happy to get to the final".
"It's not easy to do what we're doing", Atletico midfielder Saul Niguez said.
With a sixth-placed Premier League finish now nearly certain, the Europa League offered Wenger his last chance to return Arsenal to Europe's elite club competition but he will now leave with the club in their lowest position since he became manager 22 years ago. They were in unbelievable positions and they have great defenders. "We wanted to try our best to finish at the top and that adds a little bit more disappointment for us".
Koscielny might miss the World Cup
By the end of the half, Atletico had regained the ascendancy and in injury-time, they struck.
Arsenal tried to rally but struggled to break through Atletico's stout defense, creating few scoring opportunities.
The host, which wasted decent chances through Koke and Griezmann, eventually punished its opponent just before half-time - Costa holding off Hector Bellerin and finishing past David Ospina inside the area following a swift break.
Costa and Griezmann spurned chances for Atletico when they missed the target from close range.
"I am sad and very disappointed overall, very frustrated as well", Wenger told a news conference. "But we have to be cautious". We were 1-0 up with 10 minutes to go at home, and then we conceded a stupid goal.
"It was a very intense game between two teams that gave everything".
Atletico didn't have Simeone on the bench on Thursday because he was sent off in the first leg.
But however the next month plays out, the big target now for Simeone's players to is to win the Champions League in their own stadium in 12 months' time because they simply can not allow Real to lift the European Cup in their new home.
"I felt how the atmosphere was. They were with us every step of the way, they knew the side needed them".