The visitors were forced into a change after just 13 minutes, with Jordan Henderson taken off for Fabinho in the middle of the park as the skipper had picked up a knock.
After a slow opening, Liverpool drew first blood when Sadio Mane skinned Manuel Neuer to find the bottom corner with a neat finish on 26 minutes.
With six minutes remaining in the half, the Bavarians drew level after Joel Matip’s defensive clearance landed into his own net.
From the counter- attack, Neuer palmed away Mohamed Salah’s powerful left-footed strike after the Egyptian charged down the right five minutes into the second half.
With just over 20 minutes to play, Virgil Van Dijk’s towering presence put Liverpool back in front with a towering header past Neuer.
Liverpool weren’t done as Salah put the brakes on the Bayern defence to set up Mane, but Neuer outstretched leg prevented a potential third goal on 75 minutes.
There was no stopping the visitors though, as Mane leaned forth to nod past Neuer for his second of the evening on 84 minutes to seal Liverpool's spot in the last eight.