Mane caps thrilling Liverpool fightback to sink Villa
Jurgen Klopp's side were in danger of losing in the league for the first time this season after Trezeguet put the hosts ahead in the first half at Villa Park.
The late escape maintained Liverpool's six-point lead over second placed Manchester City, who also came from behind to beat Southampton 2-1.
It may only be the start of November but it was another significant hurdle overcome as Liverpool came from behind for the third successive league game and have still only dropped two points this term.
Much talk beforehand had centred on the match with no date, namely the League Cup quarter-final between these sides, which clashes directly with Liverpool's involvement in the Club World Cup in Qatar in December.
Klopp's impassioned speech about the crowded schedule may or may not be listened to but this was the first of 15 matches between now and January 2.
Influential midfielder Fabinho was rested, although this had more to do with him being on four bookings and Klopp not wanting him to get another yellow card and be suspended for the huge game with City at Anfield next weekend.
Villa started with real energy as Anwar El Ghazi tested Alisson before striker Wesley glanced a header away from team-mate Bjorn Engels.
Liverpool's obvious threat emerged as Mane headed just wide from an inviting Jordan Henderson cross and then Mohamed Salah sneaked in at the back post and forced an alert save out of Tom Heaton.
But the European Champions were stunned when they fell behind to a first goal in English football from Trezeguet.
The Egyptian volleyed in from John McGinn's free kick as the defenders waited in vain for an offside call, which did not come from either the assistant referee or VAR.
- Sting in the tale -
Roberto Firmino had the ball in the net from Mane's low cross only to see the flag up straight away.
It went to VAR where Firmino was adjudged to be offside by the narrowest possible margin, with his armpit being ruled the wrong side of the invisible line.
To complete a hat-trick of VAR calls in Villa's favour, replays backed up the decision by referee Jon Moss that Mane had dived under pressure from Frederic Guilbert to try and win a penalty for which the Senegalese was booked.
After the interval, there was immediate urgency from Liverpool as they pressed for an equaliser.
Their best opportunity came when Mane arrived at the back post to meet another delicious ball from Alexander-Arnold but rather than finding the net, his header came off Heaton's chest as the goalkeeper made himself big.
Klopp had to act and made a double change with Divock Origi and Alex Oxlade-Chamberlain thrown on, but the surprise was that Salah was one of the men to go off.
Adam Lallana had a wonderful chance to bring his side level when he arrived to meet Firmino's pull back from only six yards out but he failed to convert and at this stage nobody was expecting Liverpool to produce such a sting in the tale.
Mane's deep cross found the unmarked Robertson and the Scot headed in at the far post from close-range with four minutes left.
Then, in the fourth minute of stoppage-time, Mane glanced in Alexander-Arnold's corner to win it.