Erik ten Hag's careless side twice threw away a two-goal lead in a pulsating 3-3 draw in Istanbul on Wednesday evening to remain bottom of Group A and with their hopes of progression now firmly out of their own hands with only one more match left to play.
United will be out of the competition if FC Copenhagen spring a huge shock away at Bayern Munich later on Wednesday night, while their only hope of advancing to the knockout stages now rests with them beating Bayern - who have already qualified for the last-16 with two games to spare - at Old Trafford on December 12.
Even if they managed that impressive feat, then United would still need Copenhagen and Galatasaray to play out a draw in Denmark on the same night to sneak through as runners-up in the group.
Andre Onana endured a torrid night at a wet and windy Rams Park, where the game only went ahead after a late pitch inspection after torrential rain and storms battered Istanbul throughout the day and led to farcical conditions for the Uefa Youth League game between the clubs earlier in the afternoon.
The Cameroon goalkeeper committed two vital errors that allowed Chelsea loanee Hakim Ziyech to net a pair of free-kicks, first to reduce the deficit to 2-1 and later 3-2 as United could not hold a pair of two-goal advantages amid a hostile atmosphere.
It had been a dream start to the game for the visitors, who made four changes from Sunday's impressive 3-0 win at Everton in the Premier League as Diogo Dalot, teenager Kobbie Mainoo, Anthony Martial and the suspended Marcus Rashford made way for Aaron Wan-Bissaka, Sofyan Amrabat and the fit-again duo of Antony and Rasmus Hojlund.
Garnacho, fresh from scoring that incredible overhead kick at Goodison Park, continued his week to remember as he thundered into the roof of the net with just 11 minutes on the clock after good link-up play between Hojlund and Bruno Fernandes.
United captain Fernandes doubled that advantage just seven minutes later, rifling past Fernando Muslera with a terrific strike from 20 yards out.
However, Galatasaray received a lifeline after Onana took a fatal step to his left before a Ziyech free-kick and was then unable to shift back in time as his former Ajax team-mate curled around the wall and in.
There were hatfuls of chances at both ends in a remarkable game as the offside flag denied Mauro Icardi a leveller on the stroke of half-time, with United then going 3-1 up 10 minutes after the break when Scott McTominay slid in to side-foot home a Wan-Bissaka cross from the right.
Again that two-goal cushion proved frustratingly short-lived however, with Onana throwing a hand at Ziyech's latest free-kick after an awkward bounce and succeeding only in deflecting it into the back of the net.
That latest blunder set up a grandstand finish in Turkey, with Galatasaray getting the equaliser their efforts deserved when substitute Kerem Akturkoglu was sent through by Ziyech and took a great first touch before smashing beyond Onana - who was left limping after a collision with Icardi - at his near post.
The non-stop drama and chances continued right up until the final whistle, with Fernandes striking the post and Wilfried Zaha having a few opportunities to deal his former club a knockout blow.
Muslera also made a crucial last-gasp stop to deny fellow Uruguayan Facundo Pellistri as United could not find a fourth and final goal to leave their Champions League destiny in their own hands.