Italian champions Inter will face Barcelona in the semis after goals in the space of three minutes from captain Lautaro Martinez and Benjamin Pavard cancelled out a Harry Kane strike for Bayern, which had levelled the tie in the 52nd minute.
Eric Dier's looping header in the 76th minute earned Bayern a draw on the night in soaking and blustery Milan, but that wasn't enough for Vincent Kompany's side to continue their bid to be crowned kings of Europe for a seventh time.
Simone Inzaghi's side came through in awful conditions at the San Siro, where a combination of heavy rain and driving winds affected both teams' ability to play their best football but added uncertainty and tension to an already highly-charged occasion.
"We don't give up, we have personality and heart and intelligence. We were definitely on the back foot at times today but we showed again today that we can do great things," Martinez told Amazon Prime.
Inter are still on for a repeat of the Serie A, Champions League and Italian Cup treble won under Jose Mourinho in 2010, as they are also three points ahead of Napoli in Italy's top flight and face AC Milan in the semi-finals of the domestic cup.
Wednesday's win showed that Inter are more than capable of reaching the final of Europe's elite club competition, as they did two years ago when they were narrowly beaten by Manchester City.
"We think about winning everything at the start of every season, because if we don't then we shouldn't play this sport. That's our mentality," added Martinez.
Inter will benefit from playing the second leg of their showdown with Barca, whom they beat in the semi-finals on their way to victory in Madrid 15 years ago, at the San Siro on May 6.