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The visitors were dealt an injury blow less than 20 minutes in when top scorer Yaya Toure was forced off with what appeared to be a groin injury.
Hart was furious with his defenders when Steven Gerrard was left unmarked in the box during a corner, giving the Liverpool captain a free header, but the England keeper was able to tip the ball over.
City had a dangerous spell after the half-hour mark, as Edin Dzeko drilled the ball just wide from a David Silva pass forward, before the striker appealed for a penalty a few moments later after being brought down by Mamadou Sakho.
Philippe Coutinho had a chance to extend the home side's lead further shortly before the break, but he dragged it well wide, while Simon Mignolet had to deny efforts from Fernandinho and Silva late on.