Mohamed Salah and Sadio Mane struck in the space of five minutes near the start of the second half in an away leg played in Budapest’s Puskas Arena because of Covid-19 restrictions.
It came after the Reds had fallen to three straight defeats in the Premier League – results that have led Klopp to concede defeat in their defence of the English top-flight title.
But Tuesday’s display was far more like the dominant side of last season, with their greater experience and quality ultimately telling against an error-prone German side.
“A lot of people probably expected us to slip again because of the situation, and the boys didn’t and I’m really happy about that,” Klopp told BT Sport.
“It was the game we wanted, the game we needed. Leipzig can be a real monster – they overrun teams, they are really physical and tonight we controlled them in an exceptional way.
“For two years we’ve been really good and this year we have some problems. It’s normal that the talks start – I have no problem with that. But the boys showed what they can do.”
Liverpool had largely dominated possession and created the better chances before the break but two defensive errors and two neat finishes swung the game and the tie their way.
Salah showed superb awareness to quickly latch on to a loose back-pass before finishing neatly.
Mane was then equally alert to take advantage of a fluffed attempted clearance from Nordi Mukiele to charge through and side-foot past Peter Gulacsi.
However, they had been fortunate not to fall behind inside five minutes when Dani Olmo’s header from an Angelino cross struck the base of the post.
Having managed the game well after going 2-0 up, Klopp’s side will defend their advantage at Anfield on 10 March, when they will be firm favourites to go one round further than they managed last season.