The in-form Sadio Mané scored a brilliant opener and, although the visiting side were pegged back before the interval by Joël Matip's own goal, a towering header from Virgil van Dijk and another from Mané rewarded Liverpool's attacking intent.
"Scoring three goals here is massive and a big step for us, a big, big step", the former Borussia Dortmund manager said.
GETTY IMAGESVirgil van Dijk outjleaps the Bayern defence to head in Liverpool's second goal from James Milner's cross
Maybe we're being a bit harsh on Neuer here.More news: U.S. dollar drops amid rising sterling
'I really think this club deserves to have awareness again.
"What can I say - I can write a book about his skills, about his strengths, what a fantastic person he is", Klopp said. Maybe we were trying to play defensive and trying to score goals and maybe that was wrong and today Liverpool were better. "So young, so strong, already so mature".More news: 49ers sign Tevin Coleman to two-year deal
"In the first half we scored an outstanding goal - wow - and by half-time time everybody knew it would not be easy for Munich, and that is important information for the opponent after they were happy with a 0-0 at Anfield".
"We had better passing in the second half, we had so many good football moments in the second half and I like that we developed into the game". There are some places in world football that whatever team comes here - Manchester City, Real Madrid, Barcelona - they wouldn't expect to win it. It's not sweeter for me because it's Bayern.More news: Horner: Red Bull still way off competing with Mercedes and Ferrari
"Today we didn't play like in the Bundesliga, of course I don't know why". I wanted to win for Liverpool, not to beat Bayern. That doesn't bring me anything.