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The results leave Tottenham Hotspur top of the table on 24 points level with Liverpool.
Chelsea came from behind to beat Leeds United 3-1 at home, set on their way by a fifth goal in four days from French forward Olivier Giroud.
Edinson Cavani came off the bench to score twice for Manchester United who overturned a 2-0 half-time deficit at Southampton to win 3-2 -- a club-record eighth successive away league win moving them into the top half.
Liverpool are now unbeaten in 64 league games at Anfield setting a new club record and they were full value for the win.
Spurs move on to 14 points from seven games and are just two behind leaders Liverpool but manager Jose Mourinho cautioned against talk of them being title contenders.
Chelsea cruised to a 3-0 scoreline at winless Burnley while City edged Sheffield United 1-0 away with full-back Kyle Walker scoring against his former club.
outhampton beat Premier League leaders Everton 2-0 at St Mary's Stadium as Carlo Ancelotti's side became the final club to lose a game this season.
The win moves Leeds up to third on 10 points, albeit having played an extra game, two points behind Villa, while leaders Everton, who have 13 points, are at Southampton on Sunday.
Gareth Bale's first appearance for Tottenham Hotspur for seven years, following his loan move from Real Madrid, was eclipsed by West Ham United who scored three times late on to draw 3-3 in the London derby.
A late equaliser from Jannik Vestergaard earned Southampton a 3-3 draw away to Chelsea.