After five successive European defeats away from Anfield, Klopp's side - losing finalists last season - came good when it mattered to beat the German champions in the Allianz Arena to secure a place in the last eight.
Sadio Mane's brilliant opener was cancelled out by Joel Matip's own goal before half-time but Virgil Van Dijk's first Champions League goal for the club meant Bayern then had to score two more.
Mane converted a far-post diving header late on.
"We set the bar for LFC, for this wonderful club, that we really are back on an international landscape of top football," said Klopp. "We all think that is where the club belongs and I am really happy about the result.
"I knew we had a chance and the boys made it happen and that is brilliant, absolutely brilliant.
"There are some places in world football that if Man City, if Barca, if Real Madrid come here it is not expected that they will win, they know it is a very tough place to come.
"It makes me really proud and happy because I really think this club deserves awareness again. We are back, let's carry on. We have a lot to improve, a lot to learn but we are back and I like that."