The Senegalese was a substitute for the 4-2 win at Old Trafford.
After the final whistle, Mane refused to shake Klopp's hand.
Speaking ahead of Sunday's Premier League fixture with West Brom, Klopp indicated to Sky Sports: "We spoke and he apologised. That's clear and that is it."
Mane was back in the starting line-up for the 2-1 victory over West Brom, delivering a committed performance as Alisson's late winner kept the club's top-four hopes alive.