Mason Mount had another Chelsea effort ruled out for offside leaving the teams level after 90 minutes and forcing extra-time, past midnight local time, and Liverpool struck after just five minutes with Mane driving home after an excellent exchange with Firmino.
Liverpool, who beat Tottenham Hotspur in June's Champions League final, have now won the trophy on four occasions while Chelsea have lost their last three appearances in the match.
Klopp is now faced with the prospect of playing Andrew Lonergan at Southampton.
Meanwhile, the Reds will hope that Adrian recovers in time for the Southampton clash, with the Spaniard out with a swollen ankle.
"The Liverpool manager paused his TV interview to call out the keeper's name and referenced a scene from the hit film Rocky".
The Reds and Blues were on very different ends of very similar drubbings on the opening weekend of the 2019-20 Premier League season.
"I don't think anyone in the stadium wanted to see twice 15 minutes", Klopp said. Nobody wanted extra-time.
DeMarcus Cousins suffers another major injury
Then he suffered a torn quad muscle in the first round of the playoffs and did not return until the NBA Finals. Cousins was signed to a one-year deal in the offseason and looked poised to take over as the starting center.
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Ish Sodhi was on fire in the Global T20 Canada League and picked up eight wickets from just his first two matches. Additionally, Sodhi is a quality spinner and can make things happen even when the batsmen are under control.
After decades, UN Security Council to discuss Kashmir but behind closed doors
Warning of imminent conflict, he said: "I am telling Narendra Modi that if you try a skirmish, we will respond tenfold". Pakistan's PM Imran Khan earlier criticised India's move, warning "the world's eye is on Kashmir".
Chelsea manager Frank Lampard took positives out of his side's performance.
City played 61 games on their way to a domestic treble last season, and Liverpool must realistically plan to play at least 60 this term, unless they suffer similarly tough draws in the entry rounds of the domestic cup competitions as they landed past year. "It was a insane week but with (my team mates) it's so easy playing at the back". Twelve months ago, the Liverpool squad started the season after a much lighter World Cup workload than their rivals, but this year, the boot is on the other foot.
Jurgen Klopp has shed light on his decision to substitute Alex Oxlade-Chamberlain at half-time of Liverpool FC's victory over Chelsea FC in the Uefa Super Cup on Wednesday night.
Klopp continued: 'Before the season at the Premier League meeting they told us do not get involved with anything like that, but we are all human beings and if they get close you do not know the person and for us it is not amusing.
"It was a big fight tonight", Klopp said.
As for the opposition, Liverpool, have a few injury and fatigue concerns themselves.