Georginio Wijnaldum's early opener was quickly cancelled out by Issa Diop, with Pablo Fornals then putting the visitors ahead soon after half-time.
Mohamed Salah equalised, before Mane converted Trent Alexander-Arnold's cross to secure an 18th successive Premier League win for the Reds.
See the highlights of the clash below as our Replayed series on YouTube recaps every 2019-20 league fixture each weekday.