Erik ten Hag's side were 2-0 and 3-1 ahead in Turkey but were made to settle for a share of the spoils, with the result leaving them bottom of Group A ahead of the final matchday.
Man United are level on points (four) with third-placed Copenhagen, who face already-qualified Bayern Munich later tonight, while Galatasaray are up into second on five points.
The 20-time English champions will host Bayern in their final group fixture on December 12, while Copenhagen will welcome Galatasaray, with second spot in the section still to be decided.
There were major concerns that the match would have to be postponed due to the heavy rain in Istanbul, but UEFA were always confident that the fixture would go ahead as planned and that proved to be the case, albeit in difficult conditions.
Man United head coach Ten Hag made four changes from the weekend clash with Everton; Rasmus Hojlund, Antony, Sofyan Amrabat and Aaron Wan-Bissaka all came in, while there were familiar faces in the Galatasaray lineup in the shape of Lucas Torreira, Tanguy Ndombele, Wilfried Zaha and Hakim Ziyech.
Galatasaray had a half-chance inside the first two minutes when a fast break ended with Ziyech having a shot on goal, but Luke Shaw was across to make an excellent block, and Torreira then headed over from the resulting corner.
Man United goalkeeper Andre Onana then had to keep out a strike from Mertens in the fourth minute, before Hojlund had an effort deflected wide of the post down the other end after meeting a cross from Alejandro Garnacho during a fast start in what was an incredible atmosphere.
The Red Devils took the lead in the 11th minute of the contest through Garnacho, who found the back of the net with an excellent finish high into the roof of the net after Bruno Fernandes had picked out his teammate with a clever pass through the legs of Sacha Boey.
Galatasaray came close to levelling in the 15th minute through Torreira, but Onana made an excellent save to keep out the forward's header from a corner; there was then a check for a penalty for a handball on McTominay in the same period of play, but it was cleared by VAR.
Man United then doubled their lead in the 18th minute courtesy of a stunning strike from Fernandes, with the captain's effort flying past Fernando Muslera in the Galatasaray goal to silence the home supporters.
The home side had one back in the 29th minute through Ziyech, with the attacker's free kick finding a route into the bottom corner, and there were question marks over Onana's positioning between the sticks.
As expected, the goal changed the atmosphere inside the stadium, with Galatasaray back on the front foot, and Ziyech fired one over the crossbar in the period that followed after breaking past Garnacho.
Harry Maguire did brilliantly to deflect a strike from Zaha behind for a corner in the 35th minute, with Man United wobbling under severe pressure from the Turkish giants; McTominay then headed over down the other end, as the visitors continued to look threatening in the final third of the field.
Mauro Icardi had the ball in the back of the net in the 42nd minute, as he broke into a shooting position before firing the ball past Onana, but it was ruled offside and then confirmed by VAR, with Man United given a huge let-off late in the half, and the English side were able to enter the interval 2-1 ahead.
Galatasaray had a chance to level early in the second period when Mertens delivered a cross into Abdulkerim Bardakci, but he turned just wide of the post from close range.
Fernandes then tested Muslera with a cross in the 52nd minute which ended up being straight at the experienced goalkeeper, before Garnacho fired wide of the post in the 54th minute, with Man United threatening to score a third in Istanbul.
Man United did indeed make it 3-1 in the 55th minute, with Wan-Bissaka's cross from the right being turned into the back of the net by McTominay.
Galatasaray answered back once again in the 62nd minute, with Onana making a horror mistake, as he allowed what looked to be a harmless free kick from Ziyech to find the back of the net.
Onana managed to keep out a strike from Kerem Akturkoglu in the 65th minute, before also denying Ziyech from a wide area, but the pressure was rising, with Galatasaray putting serious pressure on the visitors every time that they broke forward.
The home side made it 3-3 in the 71st minute through Akrurkoglu, who fired a powerful effort past Onana at his near post, with the stadium erupting ahead of the final 20 minutes of action.
Garnacho just missed the post with a wonderful strike in the 76th minute, before McTominay fired one just wide of the target, with Man United coming close to a fourth on two occasions.
Facundo Pellistri then fired over the crossbar in the 82nd minute, before Onana did well to deny Zaha down the other end of the field, with the two teams still level at 3-3.
Fernandes struck one off the outside of the post down the other end, and Zaha then smashed one over the crossbar in an end-to-end period of the match.
Anthony Martial passed up a brilliant chance in the 87th minute, with the Frenchman taking too long on the ball inside the Galatasaray box, before Pellistri was somehow denied by Muslera from close range.
There were three minutes of added time in Istanbul, but an incredible contest ultimately finished level, with Man United remaining bottom of the section, while Galatasaray have moved up into second ahead of Copenhagen's clash with Bayern Munich.