The former Benfica stopper was caught in the face by a high foot from Mane, who was subsequently given a straight red card by referee Jonathan Moss.
City were not to be denied in stoppage time as Sane was afforded time and space, and unleashed a superb curling effort from 25-yards out to claim City's fifth goal.
Gabriel Jesus headed in a second in the fifth minute of added-on time and when he made it 3-0 eight minutes into the second half, it was nearly a surprise City restricted themselves to only two more.
Forget the 37th-minute red card for Sadio Mane, this was the most debatable decision.
Sergio Aguero had already put the home side 1-0 up before Sadio Mane saw red for a high boot on Brazilian goalkeeper Ederson.
"I think he's got it wrong".
Kane, last season's top scorer in the league, clipped in a second goal from close range in the first minute of the second half.
Referee Jonathan Moss deemed the challenge reckless and risky enough to merit a straight red card.
"But we controlled the game and the second goal before half-time was so important".
The 25-year-old did play for Brazil during the recent worldwide break and while Klopp agreed Coutinho could have come on at some point at City, he said it made more sense to get the player fit and ready for the rest of the season. I thought "oh my god, I hope it's not serious" but it's unlucky.