Last year’s finalists eased into the quarter-finals thanks to two goals from striker Sadio Mane and another from Dutch centre back Virgil van Dijk.
“The second half was very mature,” said German manager Klopp.
“In the moments we did play football we immediately destroyed the organisation of Bayern. It’s absolutely deserved.”
Meanwhile, treble-chasing Barca qualified to the next round on a 5-1 win aggregate against Olympique Lyon.
Liverpool’s win means there are now four Premier League sides in the last eight, following victories for Tottenham, Manchester United and Manchester City.
Barcelona, Juventus, Porto and Ajax make up the rest of the quarter-finalists. The draw takes place tomorrow at noon.