Mahrez supplied the needed pass for Jamie Vardy to put the Foxes ahead after just three minutes played at Anfield.
In the 52nd minute, Mane found Salah with a superb backheel pass as the Egypt international equalized for the Reds.
The former Roma player then netted the winner after he breached the foxes’ defence to mark his 23rd goal of the season and 17th in the league.
Salah featured for 83 minutes before making way for Georginio Wijnaldum. Mane and Mahrez were also taken off by their managers
Nigeria’s Wilfred Ndidi featured from start to finish but could not prevent the loss while Ahmed Musa was an unused substitute. Kelechi Iheanacho was not dressed for the tie.
Liverpool now have 41 points from 21 games while Leicester have 27 from the same number of games. The Merseyside outfit finished the year in the fourth spot with their visitors maintaining the eighth position.