Jurgen Klopp's men remain within six points of leaders Chelsea after seeing off the challenge of the Citizens at Anfield, with an early goal making the difference.
The world's most expensive footballer scored an excellent header a minute after Anthony Martial had equalised following Grant Leadbitter's opener at Old Trafford.
The Brazilian stepped up to twice restore the Blues' advantage, before Diego Costa swooped to put the game beyond doubt late on at Stamford Bridge.
Mauricio Pochettino's men were pegged back by Virgil van Dijk's early opener but responded in style, with Dele Alli scoring twice and Harry Kane and Son-Heung Min also netting.
Jon Walters opened the scoring at Anfield, but Jurgen Klopp's side hit back through Adam Lallana, Roberto Firmino, a Giannelli Imbula own goal and Daniel Sturridge.
The Armenian brought Old Trafford to its feet with a sublime back-heel flick as Jose Mourinho's men eventually overcame stubborn resistance to win comfortably.
The Belgium international scored a second-half penalty as Antonio Conte's men moved nine points clear at the top of the Premier League table on Boxing Day.
Arsene Wenger's side reignited their title hopes with victory over the Baggies as the France international celebrated his return to the starting line-up with the winner.