United were in desperate need of a lift after making their worst start to a league campaign in 34 years.
Rasmus Hojlund provided positive early signs with a swashbuckling run to get past Sacha Boey and cross that found Bruno Fernandes, although the captain made a mess of his attempt.
The Dane made no mistake when the roles were reversed though, as Marcus Rashford raced onto Casemiro’s pass and squared the ball for Hojlund to head home his second goal in as many Champions League matches.
The Red Devils’ lead only lasted six minutes, though, as Wilfried Zaha scored the fourth goal in five games against his former club to send his new supporters into raptures.
The winger outmuscled Diogo Dalot to reach Davinson Sanchez’s pass forward and struck an effort that hit the Portuguese right-back, cannoned off the floor and bounced over Andre Onana.
Both teams pushed to go ahead and Christian Eriksen provided Marcus Rashford with a magnificent opportunity to put his side ahead. However, the forward tried to square the ball to Fernandes but saw his pass cut out by Boey and was booed as a result.
There was soon more frustration, as Mount saw Fernando Muslera tip his strike over the crossbar before the midfielder found Hojlund, who finished clinically again but had strayed marginally offside.
Nothing could stop the dynamic Dane after Sanchez slipped though, as the striker powerfully sprinted away from Bardakci and delicately dinked the ball past Muslera.
But Galatasaray swiftly responded again, with Kerem Akturkoglu latching onto Barıs Yilmaz’s pass and coolly slotting his shot past Onana.
And the drama continued before ramping up another notch, as Onana conceded possession and Casemiro was sent off for diving in on Dries Mertens, giving away a penalty that Mauro Icardi dragged horribly wide.
Undeterred, the Argentine provided the coolest of dinked finishes to give Galatasaray their first Champions League win since 2018, while they have navigated their first 15 competitive matches this season without suffering defeat at all.
Man United, meanwhile, have lost six of their 10 matches across all competitions this season, with their long-suffering coach now potentially days from dismissal after a dark night at Old Trafford.
Flashscore Man of the Match: Rasmus Hojlund (Manchester United)
Click here to see all the stats of the match.
Bayern Munich come back to beat Copenhagen
In the night's other Group A match, second-half goals from Jamal Musiala and Mathys Tel helped Bayern Munich to a 2-1 win away at FC Copenhagen.
