It also meant United had failed to win any of their last 10 league and cup meetings with Chelsea over the past three years.
Jesse Lingaard scored a fine goal for the visitors after an hour's play but in the first minute of added time Costa coolly beat Spain team mate David De Gea to snatch a draw.
Chelsea, who lost defender Kurt Zouma with a serious looking knee injury, were left in 13th place in their unexpectedly feeble title defence. United stayed fifth, six points behind neighbours Manchester City and the Champions League places.


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