Sadio Mane scored a brilliant opener to hand Liverpool a crucial away goal, although their lead was soon cut by a Joel Matip own goal as he turned in former Arsenal winger Serge Gnabry's cross/
But in the second half Liverpool pulled clear with goals by Virgil van Dijk and another Mane finish, to send them into the final eight.
For Bayern it was a disappointing result having managed to hold Liverpool to a goalless draw at Anfield.
Liverpool become the fourth Premier League side to reach the Champions League quarter-finals, joining Manchester United, Manchester City and Tottenham Hotspur to make up half of the remaining draw.
Take a look through our gallery above to see the story of a famous night for Liverpool.