Mane provoked the Dane's dismissal and then headed the Reds into the lead before pouncing on another awful error from the Blues' goalkeeper Kepa Arrizabalaga.
Chelsea host Liverpool in the game of the weekend at Stamford Bridge.
Their first real opportunity came from a fortuitous spin of the ball, though.
Klopp has Alisson in goal against Chelsea, while Trent Alexander-Arnold and Andrew Robertson operate as the two full-backs.
Jordan Henderson's block challenge on Reece James in midfield sent it careering out towards Mohamed Salah on the right edge of the box and the Reds No.11 took the chance to carry it around the oncoming Kepa. The referee, at first, blew for a foul and showed him a yellow.
Klopp instantly called for Thiago to make his debut at half-time in place of the injured Henderson, and it took just five minutes in the second-half to make the breakthrough.
Having completed the quickfire signings of Thiago Alcantara and Diogo Jota, senior sources at Liverpool have told The Athletic that there will be no more incomings before the October 5 deadline.
Jorginho's stutter-step run up has fooled many a goalkeeper in the past, but Alisson held his ground before diving low to his left to save.
Mane again did the damage as he met Firmino's cross with a firm header into the far corner.
Mane's determined closing down of Kepa reaped rewards when he managed to block the Blues goalkeeper's attempted play out, allowing the Liverpool man to swipe home a finish on the turn.
Alisson had to repel a unsafe Tammy Abraham effort, too, but Klopp's charges saw out the remainder to make it a maximum six points in the Premier League.