Atletico will play in the showpiece match for the second time in three seasons against either city rivals Real Madrid, who beat them in the 2014 final, or Manchester City, with the second semi-final scheduled for the Bernabeu on Wednesday.
Bayern’s exit ended their hopes of claiming a treble with Atletico going through to the May 28 final at the San Siro in Milan after winning the first leg 1-0 last week.
The Bavarians went ahead with a Xabi Alonso free kick just past the half-hour and could have scored again three minutes later but Thomas Mueller’s penalty was saved by Jan Oblak.
Atletico levelled through Frenchman Antoine Griezmann’s seventh goal of this season’s competition in the 54th, meaning Bayern would need at least two more goals to progress.
Bayern’s Poland striker Robert Lewandowski set up a dramatic finale with his 74th minute goal before Atletico’s Fernando Torres had an 85th minute spot kick saved by Manuel Neuer.
But it made no difference in the end as Bayern suffered a third straight elimination by a Spanish team at this stage in their three seasons under Pep Guardiola.
“They were better, but we scored an away goal, which was important. We need to keep grounded and first of all think of the Liga before the final,” said Griezmann, whose side are level on points with leaders Barcelona in the Spanish title race.
“We’re delighted. It was difficult, a really hard game, we had a great opponent in front of us and we had to both defend and try to score. That’s what we did,” he added.