City's Bernardo Silva and Mateo Kovacic both misfired in the shootout to put Madrid in the driving seat
After the last-16 first leg ended goalless at Anfield, five-times European champions Liverpool took control in Munich, bossing the game and nullifying their German opponents.
Bayern levelled 13 minutes later against the run of play when winger Serge Gnabry beat his marker and his cutback was turned into the goal by Liverpool defender Joel Matip.
Centre back Virgil van Dijk, who had missed the goalless first leg last month, killed off any German hopes of a comeback when he rose high to head home in the 69th minute before Mane headed in his second in the 84th.