Sadio Mane's excellent chipped opener sent the Reds on their way to a first away win in the competition this season to book their place in the quarter-finals.
A Joel Matip own goal brought Bayern back into the tie before the break, only for Virgil van Dijk to restore his side's lead in the second half.
Mane then netted his second of the evening in the 84th minute, becoming his side's all-time leading away goal-scorer with seven goals.
While the German coach was particularly proud of his side's first goal, it was their defensive work he singled out as a big improvement.
"In a game like this you can never rest. It is absolutely deserved. It is so difficult to play here away at Bayern Munich," he told BT Sport.
"It is a big one. I am so proud of the boys. The first goal, I will want to watch it back like 1,000 times! Defensively it was so good. We didn't defend the way we do usually in the away games of the Champions League but today we did."
Van Dijk's goal was his first in the competition this season and, having already set up the opening goal, became the first Liverpool player to both score and assist a goal in an away Champions League knockout game since February 2007.
"Virgil van Dijk gives organisation on set-pieces and he is dangerous. He is a massive threat. The size we have and the power we have... it was a big goal," he added.
"In a game like this, there will be mistakes but you have to stay in the game. Control the moments you can control and stay in the game. The game didn't look like Bayern ever outplayed us.
"We scored the other two and could have scored another one with that chance for Mohamed Salah."