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The Cottagers appeared well on course for a first last-four appearance since 2002 when Aleksandar Mitrovic put them in front five minutes after the break.
But Fulham's hopes collapsed as the tie descended into chaos when Chris Kavanagh's decision to award the home side a penalty 18 minutes from time resulted in Willian and Mitrovic being sent off.
United capitalised on the numerical advantage as it completed a turnaround with Fernandes netting twice and Marcel Sabitzer also on target.
