The Reds quickly made amends for a slow start as Joel Matip cancelled out Douglas Luiz's opener inside six minutes.
Mane's fine header 25 minutes from time then did enough to keep the pressure on City, who have a game in hand away to Wolves on Wednesday.
However, Liverpool's narrow margin of victory keeps Pep Guardiola's men top on goal difference.
The victory also came at a cost for Jurgen Klopp's men, who lost Fabinho to a muscle injury that is likely to rule him out of both Saturday's FA Cup final and the Champions League final against Real Madrid on May 28.
With one eye on Wembley, Klopp rolled the dice with his team selection as Mohamed Salah and Thiago Alcantara were left on the bench, while there was no place in the squad for Andy Robertson.
The visitors looked like a much-changed side as Villa sprang out of the traps and caught them cold. Alisson Becker had already denied Ollie Watkins by the time Steven Gerrard's men took the lead on three minutes.
Luiz's header was parried by Alisson, but only back into the path of his Brazilian international teammate to hammer the rebound high into the net.
Villa won this fixture 7-2 last season, but Liverpool this time bounced straight back from the early blow of conceding.
The hosts did not deal with a free-kick into the box and the ball eventually broke for Virgil van Dijk to square for his centre-back partner Matip to apply the finishing touch.
Chances continued to come at both ends in a frantic first half.
Mane headed inches wide for Liverpool before Danny Ings at the other end just failed to keep his header down from Lucas Digne's inviting cross.