Rasmus Hojlund scored either side of a Wilfried Zaha equaliser, but the Turkish giants valiantly fought back and capitalised on numerous defensive errors from the hosts, with goals inside the final 20 minutes from Kerem Akturkoglu and Mauro Icardi turning the contest on its head in dramatic fashion.
Icardi missed a penalty with the score at 2-2 after Casemiro was shown his second yellow for a late challenge, but the Argentine striker made up for it by netting the match-winner just nine minutes from time to condemn Man United to their sixth defeat in 10 games across all competitions this season.
The pressure continues to mount on Red Devils boss Erik ten Hag and his side have slipped to the foot of Group A, four points behind Galatasaray in second, while Bayern Munich sit at the summit with six points after coming from behind to beat Copenhagen 2-1 in Denmark.
It was one-way traffic in the opening exchanges in terms of Man United pressure, and their persistence in the final third was eventually rewarded in the 17th minute courtesy of a brilliant close-range header from Hojlund.
Casemiro initially played Marcus Rashford through down the right flank before the Englishman volleyed a perfectly-weighted delivery towards Hojlund, whose powerful header flew past Galatasaray keeper Fernando Muslera.
However, it took just six minutes for Galatasaray to restore parity, with Zaha marking his return to his former club by silencing the Old Trafford crowd with his first Champions League goal for the Turkish giants since joining the club from Crystal Palace in the summer.
Zaha, who made just four appearances for Man United during an unsuccessful spell between 2013 and 2015, ran onto a lumped ball forward and used his body strength well to keep himself between the ball and Diogo Dalot, before his deflected strike off the Red Devils defender looped over Andre Onana into the far corner.
Man United responded well to conceding and came close to regaining the lead on the stroke of half tie, but Mason Mount's strike from around 12 yards out was blocked by teammate Hannibal Mejbri, who was unable to move out of the way as the shot travelled goalwards.
Seven minutes into the second half, Rashford was presented with a glorious chance to put Man United in front when he ran clean through on goal, and instead of taking the shot, he opted to square it to Bruno Fernandes, but the pass was short and the chance went begging to the frustration of the Old Trafford faithful.
More to follow.