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Champions City came from behind to beat Spurs at the Etihad Stadium with substitute Edin Dzeko lashing home their 88th- minute winner after Sergio Aguero had cancelled out Steven Caulker's first-half opener for the visitors.
Chelsea dropped points at Stamford Bridge where skipper John Terry scored on his return from a four-match domestic suspension before being carried off with a knee injury after an accidental collision with Luis Suarez 19 minutes later.
Suarez then headed Liverpool's equaliser 17 minutes from time.
United, who beat Aston Villa 3-2 on Saturday, have 27 points from 11 matches, followed by City on 25, Chelsea on 24, and Everton and West Bromwich Albion on 20.
In the day's other game, former Newcastle United manager Sam Allardyce and ex-Magpies skipper Kevin Nolan, who scored the winner, enjoyed their return to St James' Park as West Ham United beat Newcastle 1-0 to move up to sixth.