Jurgen Klopp's side trailed to Trezeguet's first-half goal at Villa Park, but hit back to preserve their unbeaten league record this season in dramatic fashion.
Andy Robertson equalised with four minutes left before Senegal winger Mane's stoppage-time goal gave Liverpool their 10th win from 11 league games.
The Reds, chasing their first English title since 1990, are six points ahead of second-placed City.
"It is the best possible feeling. It really is very satisfying, and the winner was from a corner kick routine that we have tried a few times," Klopp said.
"It was not because we have had a couple of comebacks in other games, you just have to keep on trying because we could have had another one today."
Pep Guardiola's champions fell behind to James Ward-Prowse's early goal at the Etihad Stadium.
But Sergio Aguero levelled for City with 20 minutes left and Kyle Walker snatched the winner in the 86th minute to keep Liverpool in their sights.
"In the first season we won the title, we got 100 points and won four or five games in the last minute; last season we won more comfortably. Okay, this is like the first year," Guardiola said.
Elsewhere, Chelsea climbed to third place as Tammy Abraham and Christian Pulisic sealed a 2-1 win at Watford.
Manchester United slumped to a 1-0 loss at Bournemouth and Arsenal were held to a disappointing 1-1 draw by Wolves, but all the focus was on the drama at the top of the table.
Firmino leads Liverpool's high-tempo title charge
At Villa Park, the hosts took a shock lead in the 21st minute when Egypt midfielder Trezeguet met John McGinn's free-kick and fired through the legs of Liverpool keeper Alisson.
Roberto Firmino thought he had equalised for Liverpool, but the Brazilian was flagged for offside from Mane's cross.
The goal was ruled out by VAR, with the Premier League offering a bizarre explanation that said Firmino's "armpit" was offside.
Liverpool's equaliser finally arrived in the 86th minute when Mane's driven ball was met by Robertson and the full-back headed in at the far post.
Then in the fourth minute of stoppage-time, Mane glanced in from Trent Alexander-Arnold's corner to complete Liverpool's incredible revival.
Struggling Southampton took the lead in the 13th minute when City keeper Ederson spilled Stuart Armstrong's shot and Ward-Prowse was on hand to slot home from close range.
Aguero struck in the 70th minute when the club's record goalscorer converted Walker's low cross.
With the momentum finally on their side, City won it in the 86th minute as Walker pounced after goalkeeper Alex McCarthy failed to deal with Angelino's cross.
Abraham, Pulisic lift 'sloppy' Chelsea into third
Ole Gunnar Solskjaer admitted Manchester United had taken a "step back" after their mini-revival came to a crashing halt as Josh King's strike gave Bournemouth the points at Dean Court.
Three consecutive wins on the road at Partizan Belgrade, Norwich and Chelsea had hinted at a turning point for United.
But Solskjaer said: "We have to win games like this. We were missing some quality at times. It is a step back for us today."
Unai Emery's problems mounted as Raul Jimenez gave Wolves a 1-1 draw as fifth-placed Arsenal surrendered the lead for the third time in a week.
The Gunners appeared to have put the issues surrounding Granit Xhaka to one side, the midfielder watching from home after swearing at supporters last week, when Pierre-Emerick Aubameyang struck in the 21st minute.
But Emery has now overseen a run of two wins from the last nine league games after Jimenez rose to header in an equaliser with 14 minutes remaining.
At Vicarage Road, Chelsea's fifth successive league win was earned thanks to Abraham's fifth minute strike and Pulisic's 55th minute tap-in.
Gerard Deulofeu's 80th minute penalty came too late to stop Chelsea recording their seventh consecutive away victory in all competitions.
Newcastle won 3-2 at West Ham, Sheffield United eased past Burnley 3-0 and Brighton beat Norwich 2-0.
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