Having won the first leg 1-0 in Spain, Atletico advanced thanks to Antoine Griezmann's crucial away goal in Munich as the tie finished 2-2 on aggregate.
Xabi Alonso and Robert Lewandowski netted for Bayern on a night when both teams missed penalties.
But it was Atletico who secured a spot in the final in Milan on May 28, where they will face either Real Madrid or Manchester City.
"They were better, but we scored an away goal, which was important," said Griezmann.
"We had to both defend and try to score. That's what we did."
Bayern and Guardiola will rue Thomas Mueller's saved first-half penalty just after Alonso had given the hosts the lead by crashing home a free-kick.
There was similar drama in the second half when Manuel Neuer saved Fernando Torres' late penalty attempt for Atletico.
After Griezmann put Atletico level in the second half, Lewandowski gave Bayern brief hope when he headed home, but the Bundesliga leaders could not find the crucial third goal.
Meanwhile, with their dreams of a treble over, Bayern are left to focus on winning a domestic double in Guardiola's swansong.
"I have given my life for this team, I have fought and given my best," said Guardiola.
"We haven't reached a Champions League final, but I am proud of this team.
"We played the football we wanted to today and all that was missing was a goal. I am sorry for all the players."
They also face Borussia Dortmund in the German Cup final in Berlin on May 21.
Having been hammered in the German media for starting the first leg without Mueller, Guardiola included the Germany star and also brought in Jerome Boateng and Franck Ribery for the return.
Simeone made just one change from the first leg with centre-back Diego Godin returning in place of Stefan Savic after recovering from a hamstring injury.
Atletico arrived in Munich having won their previous seven games and had not conceded a goal in their previous six.
His shot took a decisive deflection off Jose Maria Gimenez to wrong-foot Atletico goalkeeper Jan Oblak, and Bayern could have made it 2-0 on the night just three minutes later.
Gimenez was booked for conceding a penalty when he pulled down Javi Martinez in the area, but Oblak saved Mueller's weak effort from 12 yards.
Simeone replaced Augusto Fernandez with Yannick Carrasco at the break and moved Griezmann out wide, and the tactical switch paid dividends as the visitors struck nine minutes into the second period from their first clear chance of the game.
Bayern were left needing to score twice and they got one back when David Alaba's cross was headed back across the six-yard box by Arturo Vidal and Lewandowski nodded home his ninth Champions League goal of the season on 74 minutes.
Neuer came to Bayern's rescue by saving Torres' penalty attempt in the dying stages after the striker was brought down by Martinez, but it could not save Bayern from bowing out.
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