Atletico held a 1-0 aggregate advantage from the first leg at Vicente Calderon, but Xabi Alonso cancelled it out when his free-kick found the back of the net after a deflection off Jose Maria Gimenez.
Bayern had been relentless in their pressing and were offered the chance to lead the tie for the first time when referee Cuneyt Cakir awarded them a penalty after Gimenez's foul on Javi Martinez.
However, Thomas Muller, back in the line-up after being benched for the meeting in Madrid, was denied from the spot by Atletico goalkeeper Jan Oblak.
Muller's failure to convert proved to have a huge cost, with Griezmann racing away early in the second half and slotting a disguised shot beyond Manuel Neuer at his near post.
Robert Lewandowski's 74th-minute header made it 2-2 on aggregate, but, after Fernando Torres was kept out from the spot by Neuer after a controversial penalty decision, it was not enough to offer Pep Guardiola - who suffered a third straight semi-final exit at the Bayern helm - the chance to sign off his tenure in Bavaria with a shot at the Champions League title.
Bayern controlled possession from kick-off, but, after Lewandowski headed wide in the 12th minute, Gabi offered a reminder of the threat posed by Atletico when he forced Neuer into a routine save with a powerful drive from 25 yards.
Guardiola's side were unperturbed and Jerome Boateng – starting a second successive match after returning from three months out at the weekend – picked out Muller's diagonal run into the box, who, rather than shoot, teed up Lewandowski for a shot that was blocked by the onrushing Oblak.
Lewandowski and Philipp Lahm missed further chances as Bayern applied relentless pressure, and they finally broke through in the 31st minute.
Augusto Fernandez conceded a free kick near the edge of the box for a foul on David Alaba and Alonso drilled the resultant free-kick into the back of the net with the help of a deflection off Gimenez that wrong-footed Neuer.
Despite being rejoined in the centre of defence by Diego Godin, things got worse for Gimenez two minutes later when he was adjudged to have fouled Martinez inside the box while defending a corner, conceding a penalty.
The centre-back was shown a yellow card by the referee, but Oblak bailed him out by diving to his right and saving Muller's spot-kick, bouncing back to his feet to block Alonso's follow-up.
Franck Ribery and Diego Simeone were involved in a physical confrontation before the half-time whistle, but tempers were quickly cooled.
Simeone replaced Fernandez with Yannick Ferreira Carrasco at half-time and eight minutes later Atletico had their crucial away goal.
A brilliant pass from Torres released Griezmann on the counter-attack, and the France international guided a cool finish beyond Neuer.
Lewandowski set up a tense final 15 minutes when he powered a header into the back of the net after Arturo Vidal nodded Alaba's delivery from the left back across goal.
Torres had the chance to wrap the game up when he was sent tumbling by Martinez and the referee pointed to the spot, despite initial contact appearing to be made outside of the box.
The former AC Milan striker had his penalty saved by Neuer to the keeper's right, and Oblak produced a great stop from Douglas Costa's deflected strike with two minutes remaining to ensure Simeone's side will have another chance to claim the Champions League title, having lost to arch-rivals Real Madrid in the 2013-14 final.
Key Opta stats:
- Atletico conceded their first goal after 632 minutes.
- The six Xabi Alonso goals for FC Bayern in all competitions this season have been from outside box, five of them from direct shots.
- Jan Oblak has saved two of the three penalties conceded by Atletico in all competitions.
- Fernando Torres has provided his first assist for Atlético in Champions League and the first one provided to Antoine Griezmann in all competitions.
- Only Cristiano Ronaldo (10) has scored more goals in Champions League semi-finals than Robert Lewandowski (six).
- Pep Guardiola has now been knocked out of the Champions League at the semi-final stage in each of his last four appearances in the last four.
