The Reds led for long periods, and were twice two goals in front, only to be suffocated by two free-kicks and an expertly taken third from the hosts.
United began with marvellous composure, and powered into the lead through Alejandro Garnacho and Bruno Fernandes, but Hakim Ziyech snatched one back before half-time to give the Turks encouragement.
Scott McTominay restored the goals gap soon after the break, but another Ziyech free-kick and a smart finish from Kerem Akturkoglu dragged Galatasaray level, before a frenzied end saw both sides go close to a winner.
The result ensures we remain bottom of the group, and dependent on other results if we are to progress.