Sadio Mané and Virgil van Dijk struck late to seal Liverpool's progress to the quarter-finals.
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Sadio Mané scored twice as Liverpool brushed off their recent troubles away from home in the UEFA Champions League with a commanding performance at Bayern to reach the last eight.
Jurgen Klopp's side had lost their last four on their travels in this competition but were in the driving seat from the moment Mané capitalised on Manuel Neuer's 26th-minute error in exquisite style.
An own goal from German-born Joël Matip gave the hosts hope but Liverpool reassumed control and Virgil van Dijk's towering second-half header provided breathing space. Mané's late header made sure.
View from the stadium: Jordan Maciel
Bayern simply didn't do enough over the course of 90 minutes to get the win they needed – but for Matip's inadvertent strike, there were no clear-cut chances on Alisson Becker's goal.
Liverpool weren't brilliant, but they soaked up Bayern's pressure and were clinical at the other end to claim three away goals. Liverpool become only the second team to win here in the UEFA Champions League in the last five seasons.
Man of the match: Sadio Mané
Mané had netted eight league goals in as many appearances prior to this game, and the Senegal international continued his purple patch with two more crucial efforts in the Bavarian capital.
He showed superb composure to beat Neuer for the opener, before nodding in from close range to seal victory. Bayern's defence simply couldn't handle the 26-year-old, who proved the difference over the two legs.
4: For the third time in UEFA Champions League history four English sides will line up in the quarter-finals – no other nation has managed it. In 2007/08 and 2008/09 the quartet were Arsenal, Chelsea, Liverpool and Manchester United.
1: Van Dijk's first UEFA Champions League goal came in his 19th outing in the competition.
9: Nine of Mané's 12 UEFA Champions League strikes have come in the knockout stage. When the going gets tough, Sadio gets going.
4: Matip is just the fourth Liverpool player to score a UEFA Champions League own goal after Sami Hyypiä (Beşiktaş, 2007), John Arne Riise (Chelsea, 2008) and James Milner (Roma, 2018).
100: Neuer becomes the 38th player to achieve a century of UEFA Champions League appearances, and the sixth keeper after Iker Casillas, Gianluigi Buffon, Petr Čech, Víctor Valdés and Oliver Kahn.
15: Liverpool are through to their 15th European Cup quarter-final; they won ten of the previous 14.
7: Bayern's run of seven successive UEFA Champions League quarter-final appearances comes to an end.
What does it mean?
Liverpool and Barcelona complete the line-up for Friday's quarter-final draw, streamed live on UEFA.com from 12:00CET. One slight blip for the Reds, though, is a late booking that rules Andrew Robertson out of the first leg.
For the next few weeks Liverpool's attention will be trained on their pursuit of a first English title since 1990. There's a trip to Fulham, the visit of potential quarter-final opponents Spurs and an away match at Southampton before the first leg in the second week of April.
Bayern will seek consolation with a domestic double having hit the front in the Bundesliga for the first time since September last weekend. Dortmund are the main obstacle between them and a seventh straight title, while second-tier Heidenheim form the next hurdle in the German Cup quarter-finals.