The Red Devils were torn apart by the Dons at Stadium MK in the second round of the League Cup as two goals apiece from Will Grigg and Benik Afobe embarrassed Louis van Gaal's side.
The League One outfit punished a defensively-sloppy United side after Van Gaal had made 10 changes from the team that drew 1-1 against Sunderland on Sunday.
Van Gaal is yet to win a competitive game in charge of United and the Dutchman has tasted defeat twice in three games at the helm.
Tuesday evening's team against Karl Robinson's charges was a much-changed lineup and contained the likes of Marnick Vermijl, Andreas Pereira, Saidy Janko and Reece James.
However, Thompson, who won three European Cups with the Reds during the 1970s and 80s, thinks it was the more senior players who let the club and the youngsters down in Milton Keynes.
Thompson described the defeat as "very damaging" to the club and slammed the likes of Shinji Kagawa, Danny Welbeck and Javier Hernandez, who all started.
"I think it was very damaging," he told Sky Sports News. "And I know that Van Gaal was saying after the game that these things happen and the fans have to stay with them and believe, but that was as poor as it gets."
"Yes, he tried to flag up injuries and how many young players were playing but he had a nucleus of decent players there and it was those who let the youngsters down. They were guilty of miss-placing passes."
"A Manchester United side shouldn't go down, I don't care who they play, they shouldn't go down like that. I was watching Hernandez and he looked as though he was thinking 'why haven't I been given the night off like the rest of them'".
United travel to Burnley this weekend and new British-record signing Angel di Maria could be part of the team following his £59.7 million move from Real Madrid earlier this week.
Marcos Rojo could also make his debut after the Argentina international moved from Sporting Lisbon for around £16 million. The versatile Rojo has yet to feature due to work-permit issues.