Mane, who had helped Liverpool on their way with their second goal in the 37th minute, was visibly furious after being substituted just two minutes after teammate Mohamed Salah had failed to pass to him inside the area.
Salah attempted a solo effort rather than try to pick out the Senegalese, who was unmarked and when Mane was removed for the field he gesticulated angrily from the bench.
“He was upset, that was obvious, Sadio cannot hide his emotions, I like that but [it is] all sorted, we spoke about it,” said Klopp, who said Mane was not reacting to his substitution.
“Everything is fine, we are individuals, we are emotional, it was a situation in the game obviously, what else could happen, it was not a phone call, there was a situation in the game he was not happy about and that is then completely fine,” said the German.
While he played down the incident, Klopp also suggested he was not entirely happy with the way his forward behaved.
“Would he do it exactly the same manner again? Probably not but it happened. He didn’t say any wrong words, just looked a little bit different to how he looks usually. It’s all fine,” he said.
Meanwhile, Sheffield United came back to draw 2-2 at Chelsea, Leicester beat Bournemouth 3-1, Crystal Palace edged Aston Villa 1-0, West Ham beat Norwich 2-0, Watford drew 1-1 at Newcastle and 10-man Southampton held Manchester United 1-1 in the day’s other matches.
A bit of luck gave Liverpool the lead at Turf Moor as Alexander-Arnold’s cross deflected off Chris Wood and looped over keeper Nick Pope for an own goal.
Another Burnley mistake by Ben Mee four minutes later released Roberto Firmino to play in Sadio Mane to slot past Pope.
Firmino then added a late third as Liverpool also equalled a club record with their sixth consecutive away win.
Sergio Aguero scored either side of half-time as City stayed unbeaten by hammering Brighton at the Etihad.
Kevin De Bruyne opened the scoring after just 71 seconds and Bernardo Silva added a late fourth.
“The difference was the quality of the players,” City boss Pep Guardiola said.
“Sometimes we credit the manager for success and sometimes it is the quality of the players and today is proof of that.”
Promoted Sheffield United came from 2-0 down to stun Chelsea at Stamford Bridge.
Tammy Abraham’s brace seemingly put the hosts in control but Callum Robinson got United back into it early in the second half.
Robinson then had a hand in the 89th-minute equalizer as his cross was deflected in by Kurt Zouma for an own goal.
“Coming to Stamford Bridge after just coming up, it was all new to us but we have to believe,” Robinson said.
“We have proved we can play against the best.”
The collapse leaves new Chelsea manager Frank Lampard still seeking his first home win.
Leicester remain unbeaten after Jamie Vardy’s double helped them beat Bournemouth and move up to third place.
Crystal Palace are fourth after Jordan Ayew’s goal beat Aston Villa, who saw Trezequet sent off and had a late equaliser ruled out for a dive by Jack Grealish in the build-up.
Goals from Sebastien Haller and Andriy Yarmolenko lifted West Ham past Norwich into fifth, while Will Hughes’ early score helped winless Watford earn their first point at Newcastle.
Jannik Vestergaard’s first EPL goal gave 10-man Southampton a home draw with Manchester United.
The Dane cancelled out Daniel James’ early opener and the Saints then held on after Kevin Danso was dismissed for a second yellow card.
Wolverhampton Wanderers visit Everton and Arsenal host Tottenham Hotspur in the north London derby in Sunday’s games.