The German appreciated the sentiment but called on fans to hold off until the final whistle in the future, reminding them that games can turn if players switch off and lose concentration.
After Liverpool took a 4-1 lead in their Premier League opener against Arsenal, the travelling supporters serenaded Klopp only for the hosts to claw back to 4-3.
"When I speak about this, I have to say (to the fans) ... please don't sing my name before the game is decided," Klopp told reporters on Saturday.
"Immediately when the Kop started singing 'Juergen Klopp', (Leicester) were alone in front of (Liverpool keeper) Simon Mignolet.
"It's a little bit like celebrating a penalty before you've scored," he added. "I don't play, but if they like me, it's nice, I like them all, but please don't sing my name before the game is decided.
"It was the same at Arsenal. It's something that depends on each other, it's nice but not necessary. It would be really nice if they could stop this."
Saturday's match was Liverpool's first at home this season after redevelopment work on the Main Stand at Anfield forced them to play their opening three games on the road.
"We need to have atmospheres like this, not only when it's the champions and we score four goals," Klopp said after his side moved fifth on seven points.
"What we have to do in the season (is) to create our own atmosphere for us and nobody else."
Liverpool continue their campaign away at Chelsea on Friday.
(Reporting by Ed Dove; Editing by Toby Davis)
