Manchester United on brink of Champions League exit after Copenhagen collapse
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Rasmus Hojlund looked to have come back to haunt his former club by scoring twice inside half an hour.
But Rashford's sending-off swung the destiny of the match and possibly sealed United's fate with a third defeat in four games in Group A.
Mohamed Elyounoussi and Diogo Goncalves levelled for Copenhagen before half-time.
Bruno Fernandes' penalty briefly put United back in front, but two goals in the final seven minutes from Lukas Lerager and 17-year-old Roony Bardghji handed Copenhagen their first Champions League win since 2016.