Andreas Christensen was sent off just before half-time for denying Mane a clear goal-scoring opportunity, in an incident that proved to be the decisive moment of the game.
Liverpool raced outof the blocks in the second half, and were 2-0 up in no time, as Mane punished Chelsea.
The Blues had an opportunity to get a foot back in the game when Thiago Alcantara conceded a penalty on his Liverpool debut, but Alisson Becker saved Jorginho's penalty.
Here's how each player fared for both teams.
Chelsea Player Ratings
Kepa Arrizabalaga - 2/10
Another day, another awful error from Kepa. He played the ball straight to the onrushing Mane to enable the Reds to make it 2-0, and put the game beyond Chelsea. He did make a couple of decent saves towards the end of the game, but that mistake was fatal to Chelsea's chances in this game.
Reece James - 7/10
For most of that game, James went toe-to-toe with Mane, and came out level pegging. However, for Liverpool's first goal, the young Englishman was outsmarted by Mane's movement, as he lost track of the Senegalese and let him have a free header.
Andreas Christensen - 1/10
He was first caught napping by Mane's run across him, and then compounded that error by jumping on the Senegalese and bringing him down. It was a clear red card, and left Chelsea with next to no chance of winning the game with 10 men against 11.
Kurt Zouma - 6/10
He was terrific at the back for most of the game, especially when Chelsea were defending set-pieces. However, he could not really get his passing on point, as Liverpool's press drew errors out of Zouma far too often.
Marcos Alonso - 5/10
Salah had a great time on the flank up against Alonso, even if he barely managed to provide any end-product. The Spaniard didn't get close enough to the Egyptian, and allowed him to turn far too easily.
Jorginho - 3/10
The Italian's miserable afternoon was compounded by the missed penalty, as the Reds midfield pressure really got to him. He couldn't reallyimpose himself on the game with his controlling of possession from midfield.
N'Golo Kante - 6/10
You could not question the desire or work-rate from Kante, but especially in the second half, he was just left chasing shadows.
Mateo Kovacic - 5/10
Kovacic was generally sitting a little too deep, and just recycling possession. His best work in a Chelsea shirt has generally come as a ball-carrier, and he didn't do that in this game.
Mason Mount - 5/10
Mount was a peripheral figure in the game. In the second half, he was non-existent, apart from one moment, when a dipping shot just flew over the bar.
Timo Werner - 6/10
Starting on the left, Werner had a few bright moments, when he ran at the Liverpool defence with menace, but Fabinho had a great game, easily stopping him each time. Werner also won the penalty that Jorginho missed.
Kai Havertz - 4/10
A real non-entity in this game, Havertz could not get any change out of Van Dijk or Fabinho. He was taken off at half-time.
Fikayo Tomori - 6/10
Replacing Christensen, he did look a lot better than the Dane, and was commanding especially in defending crosses.
Tammy Abraham - 5/10
He had a clear chance to score, but his shot was well saved by Alisson.
Ross Barkley - 4/10
After replacing Kovacic, he was chasing shadows in midfield.