Mane was replaced by James Milner after N'Golo Kante hauled Chelsea back into the game with a mazy dribble and exquisite finish in the 71st minute.
Liverpool, who had led 2-0 at half-time thanks to goals from Trent Alexander-Arnold and Roberto Firmino, survived numerous late scares as Frank Lampard's men pushed for an equaliser that never arrived.
That win leaves the Reds sitting pretty at the top of the Premier League table, leading Manchester City by five points after starting the season with six victories in as many outings.
Klopp told Sky Sports that Mane was "okay" after the final whistle, before explaining to reporters the reasons for replacing the forward.
Jürgen Klopp provides an update on Sadio Mane... https://t.co/crt1F8eHcZ— Liverpool FC (@LFC) September 22, 2019
"It was a knock on the bone, around the knee, and a dead leg," said Klopp. "That's an awful combination.
"He tried everything. I think you saw there was one counter situation when he really tried and I would say on a non-dead-leg day he will get that ball, but no chance.
"Then we say, 'come on, now we have to do it' and changed. That's it."
Liverpool are next in action on Wednesday, as they face MK Dons in the Carabao Cup.
Their next Premier League fixture sees them travel to Sheffield United in the early kick-off on Saturday.