The Red Devils looked set to retake the lead against their Turkish opponents in the Champions League at Old Trafford but Rashford was guilty of lacking composure.
But Rashford fluffed up a massive opportunity when put through one-on-one in the 53rd minute.
Christian Eriksen expertly found the run of the England international who arrowed his run in from the right flank.
But as he approached the onrushing Galatasaray goalkeeper Fernando Muslera, he dithered at what to do.
He decided against taking a shot, instead trying to play in with Bruno Fernandes with a squared shot.
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However his pass was under-hit, with right-back Sacha Boey able to clear the danger.
The fans in attendance couldn't believe it, with the camera cutting to Ten Hag, who was equally as flabbergasted.
It proved to be a costly moment for Man United as they fell to a surprise 3-2 defeat.
Rasmus Hojlund put Ten Hag's men ahead with a bullet header before former Red Devil Wilfried Zaha levelled on 23 minutes.
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Shortly after Rashford's glorious opportunity, Man United took the lead again through Hojlund's delicate finish as he capitalised on ex-Tottenham defender Davinson Sanchez's slip.
But Man United then collapsed in the final 20 minutes of the 90 with Muhammed Kerem Akturkoglu slotting in a second equaliser for Galatasaray.
Andre Onana's dreadful pass was intercepted by Dries Mertens, and in an attempt to make a last-ditch challenge, Casemiro was sent off for bringing down Mertens inside the box.
Mauro Icardi missed the resulting spot-kick but made no mistake on 81 minutes when he was played through by Davinson Sanchez, dinking the ball over a wayward Onana to give the travelling fans a night to remember.