The Senegal forward's excellent 26th-minute opener gave Jurgen Klopp's side a priceless away goal but, in a similar vein to the goalless first leg at Anfield, the action was fitful and error-strewn for the most part.
Bayern were level at the break thanks to Joel Matip's own goal but Niko Kovac's side were never able to mount a sustained victory push.
A typically dominant Virgil van Dijk powered in to leave a forlorn Bayern with a mountain to climb and Mane administered the knockout blow six minutes from time.
Fabinho, a surprise omission from the Liverpool starting XI, was soon pressed into action when Jordan Henderson suffered an early ankle injury.
The visitors settled well despite that setback and - moments after Roberto Firmino flashed a snapshot wide - Mane brilliantly brought down Van Dijk's raking ball, spun a needlessly marauding Manuel Neuer and chipped in a sublime finish.
Liverpool's lead lasted until six minutes before half-time when Serge Gnabry darted in behind Andy Robertson and Matip put through his own goal with Robert Lewandowski lurking.
Neuer dealt unconvincingly with a well-hit Mohamed Salah drive that was straight at him in the 50th minute, with Bayern struggling for rhythm.
Gnabry remained a reliable outlet – not to mention a menace for Robertson – and he whipped a dangerous ball between the goalmouth and a sliding Lewandowski as the hour passed.
However, the Poland striker would like to have given Van Dijk far more work before the defender rose at the back post to power home James Milner's 69th-minute corner.
Bayern were finished and Mane added late gloss on the end of Salah's floated cross.
What does it mean? Premier League era begins as Bayern's ends
Bayern last fell at this stage of the competition in 2011, going on to win the Champions League two years later. Four semi-finals in the subsequent five seasons followed but age has caught up with a fine side and a revamp is necessary, with the likes of Gnabry and Joshua Kimmich – badly missed through suspension in the second leg – set to have key roles.
The all-Bundesliga final against Borussia Dortmund was an emphatic statement, 12 months before Germany lifted the World Cup. Club and national team success do not necessarily intertwine, but Liverpool joining Manchester City, Manchester United and Tottenham in the quarter-finals should cheer those who revelled in Gareth Southgate's England reaching the semi-finals of Russia 2018.
Van Dijk underlines player-of-the-season credentials
A game-breaking contribution from the Netherlands centre-back did nothing to explain his importance to Liverpool that the tangible sense of panic surrounding his suspension from the first leg had not done already. Van Dijk was typically colossal here, with Lewandowski feeding off scraps as his marker helped himself to an assist and a memorable goal.
Watch him defend... and we watch him score! pic.twitter.com/OJiVypz1IB— Liverpool FC (@LFC) March 13, 2019
Neuer decline sad to witness
Throughout this golden period for Bayern, Neuer has rightly been considered the premier goalkeeper in world football. However, injuries have taken an awful toll on the 32-year-old, whose muddled decision-making led to Mane's opener. He pawed behind for the corner that led to Van Dijk's winner – the lack of authority inside his own six-yard box a sorry sight.
Key Opta facts
- Liverpool's progression means four English sides will play in the quarter-finals for the first time since 2008-09 (Arsenal, Chelsea, Liverpool and Manchester United).
- Since the start of last season, no player has provided more Champions League assists than Milner (10).
- Mane's brace means he has now scored seven away goals in the Champions League for Liverpool, more than any other player for the club in the history of the competition (including the European Cup).
- Matip is only the fourth Liverpool player to score an own goal in the Champions League after Sami Hyypia (2007), John Arne Riise (2008) and Milner (2018).
- Neuer became just the sixth goalkeeper to play in 100 Champions League matches after Iker Casillas, Gianluigi Buffon, Petr Cech, Victor Valdes and Oliver Kahn.
Liverpool will return to the top of the Premier League on Sunday if they beat lowly Fulham, with Manchester City in FA Cup action a day earlier at Swansea. Bayern entertain Mainz as they remain neck-and-neck with Borussia Dortmund.