Newcastle had led three times but United pegged them back each time and then won it through Javier Hernandez in the last minute.
That meant they move further clear of City, who lost 1-0 at Sunderland for the second season in a row.
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.