Joel Matip was a centre-half struggling to earn Klopp's trust prior to Van Dijk's arrival, but he is now a supremely confident central defensive partner for the main man, and his own-goal against Bayern, when he could either divert the ball into his own net or leave it for Robert Lewandowski to score, should not take the shine off his performance.
This is the first time since 2011 that Bayern have failed to make the quarter-finals in Europe's top club competition.
Rather than panic, however, Liverpool controlled proceedings in the second half, with Virgil van Dijk powering home a header and Mane killing the tie off late on.
"It's a big, big step for us". We have set a mark that this wonderful club is really at the top of the landscape of European football.
"Today we didn't play like in the Bundesliga, of course I don't know why". I'm really happy about the result. He also scored a headed goal, Liverpool's crucial second, prompting Klopp to highlight his vast value to this team.
"Sadio is on fire".
GETTY IMAGESVirgil van Dijk outjleaps the Bayern defence to head in Liverpool's second goal from James Milner's cross
The Senegal global scored twice at the Allianz, his first a wonderful turn to deceive Germany goalkeeper Manuel Neuer which team-mate James Milner described as "ridiculous", to take his tally to 10 in as many matches.
Jurgen Klopp praised Virgil van Dijk's impact on Liverpool's season and feels the best is yet to come as the club's fight on two fronts continues.
He told Viasport: "In both games we played too defensive, we didn't try to push forward to create [attacking] situations, also in the first game, and I think today we didn't risk so much".
"Even the boss is saying, why the fuck we ain't got a decent song for Mane yet", added another.
Niko Kovac says his Bayern Munich side now have to concentrate on winning a German league and cup double after their Uefa Champions League exit at the hands of Liverpool.
Information from Reuters was used in this report.
‘Avengers: Endgame’ Trailer: Say Hello To Brie Larson’s Captain Marvel, New Poster
Marvel also treated fans to a look at the film's new poster, featuring a full roster of superheroes, including Captain Marvel . The trailer eventually goes on to show how the Avengers are grieving as they recount the aftermath of the snap.