The Reds manager has conceded the defence of their Premier League title is over after three successive defeats but their European prospects are still in full swing after they seized control of their last-16 tie.
Second-half goals from Mohamed Salah and Sadio Mane in the first leg at the Puskas Arena – with the tie held in the Hungarian capital because of travel restrictions in Germany – helped Liverpool return to winning ways.
Klopp feels many would have been tipping Leipzig to prevail but he insists his side’s recent displays have not been as bad as their results suggest, even if he recognises this outcome will not instantly restore their confidence.
“We are not children anymore, so we don’t think everything is fine because we won one game,” he said.
“It is much more important that we played good football again, and not for the first time in the last few weeks.
“I understand and respect that nobody is really happy with the results, but the football we played in dominating City in a way that doesn’t happen often, it was the same against Leicester, we should not forget that.
“Here, the result came with it. We kept a clean sheet, we scored two goals, we forced the opponent to make the mistakes we made in the last few weeks.
“I think people expected us to slip tonight. I’m pretty sure a lot of people thought ‘that will be really tricky for Liverpool’. And it was, but we did it.
“We worked really hard again, there were top individual performances but really we worked together. The majority of the time, we had control of the game, which was very important and doesn’t happen a lot against Leipzig.”
Leipzig started brightly as Dani Olmo’s diving header came back off the inside of a post, but Liverpool began to assert themselves thereafter, with the Bundesliga side looking susceptible because of their high defensive line.
Both Liverpool goals came via Leipzig errors, with Salah latching on to a woeful back pass by Marcel Sabitzer to slot home in the 53rd minute, and Mane doing likewise before the hour after Nordi Mukiele’s slip-up.
However, they had Alisson Becker, who has made a few high-profile mistakes recently, to thank for keeping their clean sheet in tact just after half-time as he saved marvellously when Christopher Nkunku had been put clean through.
“Of course a clean sheet helps, as you can imagine,” Klopp added.
“Ali will sleep really well in the plane on the way home! As always, he helped us a lot.”
Substitute Hwang Hee-chan lifted over Alisson but wide of the goal in the dying seconds as Liverpool kept a clean sheet for only the second time in 11 matches, against a side who eliminated Manchester United to reach this stage.
Leipzig, who reached last season’s semi-finals and have beaten Paris St Germain and United at home this season, looked out-of-sorts at times but head coach Julian Nagelsmann feels they can count themselves unlucky not to score.
“The big difference of the day was they scored on their two occasions. We had three big chances and just missed them, we weren’t clinical enough,” he said.
“We did well over the 90 minutes. We created a lot of big chances and at the end you have to score.”
Leipzig face an uphill task to overturn the deficit in the second leg- scheduled to be held at Anfield on March 10 – but Nagelsmann is not throwing in the towel.
“The condition is not easy for us ahead of the second leg but it’s not impossible,” he added.
“We will give our everything to make it into the next round.”