Centre back Virgil van Dijk, who had missed the 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.
Although they won't say so publicly, behind closed doors Barcelona players are quite sure about who they would rather face in the quarter finals of the Champions League, after seeing off Olympique Lyon this week.
The 30-year-old is quoted by the Liverpool Echo as saying: "In both games we played too defensive".
At one point the Liverpool legend turned into the interviewer as he asked Van Dijk if beating Bayern in the Champions League will help their Premier League title bid.
Mane's goal gave them the edge but instead adding to their lead when they had the upper hand, Matip's own goal let Bayern back into the game against the run of play.
His record comes after an awesome campaign last season in which he scored 10 Champions League goals, including one in the final against Real Madrid.
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It was a result that left Klopp purring with satisfaction at a mature performance which, he said, reaffirmed the club's... Liverpool got precisely what they paid for: the best.
In three ties between Premier League and Bundesliga opposition, the English clubs won 17-3 on aggregate, which will be regarded as a humiliating outcome by Germany's top teams. "Whether teams take notice is not really for us to worry about but they take notice after they have played us and we have put them out".
His quality shined even more in contrast to Bayern's defensive pairing of Mats Hummels and Niklas Sule.
"Virgil gives organisation on set pieces, and he is risky", Klopp said with some understatement. He was so strong, so important to us, but he can play better than he did [on Wednesday].
"There's going to be many times in the Champions League you'll come up against world-class strikers, and he's the player you need to put them to sleep".