Jurgen Klopp’s men calmed fears over their staying power following a recent dip in form as they chase down their first top-flight title since 1990.
Defending champions City finally broke the resistance of West Ham at the Etihad Stadium in the second half to stay just one point behind Liverpool, while third-placed Tottenham slipped to a damaging 2-0 defeat at Chelsea.
Arsenal swatted aside a limp Bournemouth team 5-1 to remain in fourth place, one point clear of Manchester United, with Romelu Lukaku taking advantage of the absence of Marcus Rashford to score twice in a 3-1 win at Crystal Palace.
Liverpool came into their match after three draws in four league matches but Mane settled their nerves early, heading home a cross from Trent Alexander-Arnold before the pair teamed up again, with the Senegalese forward producing an audacious backheel strike.
Divock Origi extended the lead midway through the second period and defender Virgil van Dijk netted twice late on to complete the rout.
“The boys showed how much they enjoy the game and the situation,” a delighted Klopp told the BBC.
“It was an important sign. You know what was written and said about us. It is of course not right. These boys are outstanding and today they showed so much passion.”