Chelsea were reduced to 10 men after Andreas Christensen was sent off for a professional foul on Mane after 45 minutes and
There was a lack of clear cut chances in the first half with Chelsea looking to play on the counter attack and allowing Liverpool to control possession.
Just before halftime, Mane was through on goal when Christensen brought him down. The referee initially brandished a yellow card but Paul Tierney hanged it to a red after checking the VAR screen.
The visitors took the lead five minutes after the break when they produced a fine move which resulted in Mane heading in from Roberto Firmino's cross.
Mane grabbed his second four minutes later when he pounced on Kepa Arrizabalaga's mistake and scored into an empty net. Kepa's lazy pass to Kurt Zouma was easily intercepted by Mane who had the simplest of tasks to roll the ball home. Mane could also have bagged a hat-trick but his deflected effort late on was well-saved by the Spanish keeper.
Odgen: Kepa now a 'lost cause' for Chelsea
ESPN FC's Mark Ogden discusses Frank Lampard's comments after Kepa's error vs. Liverpool.
"It's always a tough game here. I liked the first half. It was two teams that wanted to play, create, defend and make things happen," Liverpool manager Jurgen Klopp said after the match.
"They were the dominant side without goals but then a massive game-changer in the last seconds which was good for us. We had to adapt to that and we did."
Chelsea manager Frank Lampard was diplomatic in his response about Kepa's poor performance amid reports that Rennes keeper Edouard Mendy will be the next major addition for the Blues.
"A keeper needs confidence, it was a clear mistake, he has accepted that but we have to have joint responsibility, these things do happen," Lampard, who thought his side's display was better than in the win over Brighton last week, said.
Lampard did confirm that Willy Caballero would start between the sticks in Wednesday's Carabao Cup match against Barnsley, but that was always his plan.
On Christensen's booking, Lampard said: "The red card changes the face of the game.
"I think it could be given or not. I was perhaps surprised it was given as a yellow first. If it was given as a red straight away, you continue with that."
"We decided to bring Thiago on early because it was 11 vs. 10 and a player like him can command the rhythm. It was the perfect start for him and the boys helped him massively," Klopp said.