- Man Utd
The Senegalese forward slotted in his second as Liverpool completely outclassed their opponents, with Roberto Firmino adding his name to the scoresheet with 20 minutes to play, before Mane's searing strike completed his first European hat-trick.
Real host PSG at the Santiago Bernabeu on Wednesday in the first leg of their last-16 tie, with the return set for March 6 in the French capital.
Liverpool moved back above Tottenham Hotspur into third and to within two points of bitter rivals Manchester United in second.
Liverpool remain in third place, while Tottenham Hotspur are two points behind Jurgen Klopp's side in fifth.
Just five points now separate Manchester United in second from Tottenham Hotspur in fifth.
Arsenal goalkeeper Cech's miscued attempt to clear allowed Jordan Ayew to slot home Swansea's second goal at the Liberty Stadium.
Jay Rodriguez scored twice as West Brom pulled off a thrilling 3-2 victory at Liverpool in the fourth round of the FA Cup.
It was a rare chastening experience this season for City, but shouldn't prove damaging to their title bid as they remain 15 points clear of second-placed Manchester United, who host Stoke on Monday.
Coutinho signed a five-year contract to end a long-running transfer saga on Monday with Mestre revealing that Liverpool had originally asked for 200 million euros.
Philippe Coutinho finally got his wish on Saturday when Barcelona agreed to sign the Brazilian star from Liverpool for 160 million euros ($192 million) in the third richest deal of all time.
According to reports in Spanish media, Barcelona and Liverpool have agreed a deal in principle worth up to 160 million euros for Philippe Coutinho to move to the La Liga giants this month.
Liverpool had led 1-0 at the break after James Milner converted from the penalty spot but Everton looked to have secured a replay when Gylfi Sigurdsson equalised midway through the second half.
The Liverpool star held off competition from Pierre-Emerick Aubameyang and Sadio Mane to take home the prize.
Brazilian international forward Coutinho has been linked with a 150 million euro (£134 million; $180 million) switch to the Spanish giants.
The Clarets thought they had rescued a point when they equalised on the 87th minute, but Ragnar Klavan met Dejan Lovren's header in the final moments.
Mohamed Salah was again instrumental for Liverpool as he took his tally to 17 Premier League goals this season.
Reds boss Jurgen Klopp has remained determined to sign Van Dijk as he tries to shore up a leaky defence that has fatally damaged Liverpool's Premier League title ambitions.
Fourth-placed Liverpool remain one point above fifth-placed Arsenal after a Christmas cracker that underlined the qualities and problems with two teams.