After a goalless draw in the first leg of their last-16 tie, Liverpool beat Bayern 3-1 in the Allianz Arena to advance to the quarter-finals.
Sadio Mane scored twice and Virgil van Dijk also netted as Klopp's side produced a controlled display to become the fourth Premier League team in the last eight.
"Scoring three here is really difficult, that's massive, and a big step for us," said Klopp.
"We will see what we can do with it, but it's a fantastic sign. I love it. We set a mark for LFC with this wonderful club that we really are back on the landscape of top international football.
"We all think that's where this club belongs and tonight we proved it at least a bit. I am really happy about the result and the fact we are through.
"There are some places in world football where whatever team comes, whether it's Manchester City, Barcelona, Real Madrid, they know it is a very tough place.
"It's not about playing your best free-flowing football, that's not possible against this quality because you cannot use your speed really because the other team is quick as well. It's not a massive advantage. You have to find different ways and tonight we found them.
"That makes me really proud and happy because I really think this club deserves the awareness again. We are back, it's so good. We have a lot to improve and develop and a lot to learn but we are back and I like that fact."
Klopp was also full of praise for defender Van Dijk, who produced another commanding display to help keep Bayern at bay.
"I can write a book about his skills, his strengths, how much I like him, what a fantastic person he is. He's so young and already so mature, so strong; all good.
"Virg knows that he can play better than he did tonight, we can play better than we did tonight. But especially in the second half, we played as good as necessary and possible in a game like this.
"It's so important to stay in the game and you are stable and have answers to the questions and can change things and use different tools.
"We had so many good football moments in the second half especially. It's a difficult place to come and not a lot of teams have an away win at Munich on their cards. We have that now and that's cool."