Sadio Mané scored twice in a superb individual display while the equally impressive Virgil van Dijk also scored in Liverpool's first European away win of the season, and arguably their finest under Klopp.
"Scoring three here is really hard - actually we scored four because we scored the other one as well!" he told a post-match news conference.
GETTY IMAGESVirgil van Dijk outjleaps the Bayern defence to head in Liverpool's second goal from James Milner's cross
'We will see what we can do with it but it's still fantastic for us that LFC as a club, we really are back for the last games of global top football. That makes me really proud and happy because I think this club deserves recognition and awareness again.
'We all feel that's where this club belongs and we proved it at least a little bit.
Van Dijk sat out the first leg through suspension and, once again, proved an influential presence on his return. What can I say?More news: UK, EU announce change to Brexit deal ahead of key vote
"It was really good".
"Tonight we played - especially the second half - as good as necessary and as possible. It is so important to stay in the game at times and answer all the questions". There are some places in world football that whatever team comes here - Manchester City, Real Madrid, Barcelona - they wouldn't expect to win it.
"I'm not like this". It's not sweeter for me because it's Bayern.More news: Android Q Beta Ready for Pixel Phones
"A lot of teams struggle here to get a result, but we still did it".More news: Mahatma Gandhi wanted Congress disbanded, claims PM Modi
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