Three-time European champions United needed a win in Turkey, against opponents who beat them at Old Trafford last month, in order to keep qualification for the knockout rounds in their hands.
Erik ten Hag's side appeared in control when Alejandro Garnacho's early opener was followed by a stunning strike by Bruno Fernandes on 18 minutes.
Hakim Ziyech pulled one back with a free-kick, only for Scott McTominay to restore the visitors' two-goal advantage early in the second half.
Yet Galatasaray were again given hope when another Ziyech free-kick was fumbled into the net by United goalkeeper Andre Onana, and the hosts equalised thanks to a brilliant 71st-minute strike by Kerem Akturkoglu.
United have now conceded 14 goals in five European matches and this result, combined with FC Copenhagen's 0-0 draw away to already-qualified Bayern Munich, leaves the Premier League side bottom of Group A with one game remaining.
In order to go through they must now beat Bayern at home next month and hope Copenhagen and Galatasaray draw.
"We are going in the right direction so I know we will be successful in the long-term but if you want to stay in the Champions League you need to win these games," said Ten Hag.
Gunners hit six
Arsenal's form in Europe this season has been in stark contrast to that of United, and the Gunners romped into the last 16 with a 6-0 demolition of Lens in London.
Mikel Arteta's side were five goals up at the break with Kai Havertz, Gabriel Jesus, Bukayo Saka, Gabriel Martinelli and Martin Odegaard all netting.
Jorginho completed the scoring from a late penalty as Arsenal gained revenge for their 2-1 defeat by Lens in France last month.
"Today was the day to get it done. We did it in a beautiful way," Arteta told broadcaster TNT Sports.
Arsenal have scored 12 goals without reply in winning their three home matches in Group B and have now wrapped up top spot ahead of PSV, who are also through.
The Dutch side, European Cup winners in 1988, produced a superb comeback to triumph 3-2 away to Sevilla and put themselves out of sight in second place.
Goals by Sergio Ramos and Youssef En-Nesyri had Sevilla 2-0 up, but Lucas Ocampos was sent off for the hosts just before substitute Ismael Saibari, the Spanish-born Moroccan international, pulled one back for PSV.
They equalised through a Nemanja Gudelj own goal and won it in stoppage time through United States forward Ricardo Pepi.
Sevilla now have to win at Lens next month to salvage a place in the Europa League, the competition they won last season.
Bellingham scores in Real win
Real Madrid were already through to the last 16 before hosting Napoli and they clinched top spot in Group C with a 4-2 victory at the Santiago Bernabeu.
Giovanni Simeone, son of Atletico Madrid coach Diego, gave Napoli an early lead, but Rodrygo equalised immediately, continuing his fine form with a superb strike for his seventh goal in his last seven games.
Jude Bellingham put the hosts ahead with his 15th goal of the season in all competitions only for Andre-Frank Zambo Anguissa to get Napoli back level just after the restart.
However, teenage substitute Nico Paz made it 3-2 for Real with a long-range strike that evaded the goalkeeper's grasp, and Bellingham set up Joselu to wrap up the win late on.
That result prevented Napoli from clinching qualification as they could still be caught by Braga, who drew 1-1 at home to Union Berlin in Portugal despite having Sikou Niakate sent off in the first half.
Robin Gosens fired Union in front but Alvaro Djalo levelled for the hosts after the break.
Real Sociedad and Inter Milan were both already through from Group D and their meeting next month will decide who tops the section.
On Wednesday, a much-changed Inter side recovered from three goals down to draw 3-3 with Benfica in Portugal after Joao Mario scored a first-half hat-trick against his old club.
Marko Arnautovic and Davide Frattesi pulled goals back for Inter before Alexis Sanchez levelled from the penalty spot, and Benfica were reduced to 10 men late on when Antonio Silva was sent off.
Real Sociedad drew 0-0 with Red Bull Salzburg.
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