They had entered the second leg against their Bundesliga opponents short on confidence as a result of a run in the Premier League that has seen them lose six straight Anfield games.
But they carved through Leipzig and might have scored six or seven against the Germans were they more clinical in both halves.
Julian Nagelsmann’s men only gave Alisson one real save to make all match although Alexander Sorloth hit the crossbar with a header after coming off the bench.
And Klopp saluted the performance of Fabinho in holding midfield as the Brazilian returned to his favoured role for the first time since October 17, having spent all of his appearances since then at centre-back.
That steeliness helped Liverpool to book their spot in next Friday’s last eight draw, with Paris Saint-Germain, Porto and Borussia Dortmund all also guaranteed to be in the pot.
Klopp told BT Sport of Fabinho: “I told him on the way downstairs, ‘You like the position six more then?’ He said, ‘Yeah.’ He couldn’t show it more clear!
“We all want Fab on the six. Tonight he played there and the two centre-halves played an incredible game, to be honest. The whole last line played a really good game.
“We had to press them. We defended deep really well, not only in Champions League but especially. Everybody had a really good game.
“The three in midfield worked really well together. Three up front. Diogo [Jota] as long as he could. It was a sensible decision to not let play him 90.
The German coach added of his side’s bright performance: “Yeah good. Created a lot of chances. That’s us – first half a lot of chances but didn’t score.
And Klopp admits his side cannot afford to wallow and must use the confidence gained of their European progress to rediscover winning momentum domestically too.
“We have to [use this],” he said. “Last time we played in the Champions League it was a long trip. We arrive tomorrow 5 o’clock or so.
“We play only on Monday so that’s OK. We have to make ourselves ready.
“If we play like tonight it will be not easy for Wolves. We want to win football games. We feel it, the first time in a while, and it makes a massive difference.”