Sir Alex Ferguson's men emerged victorious thanks to an early goal from midfielder Michael Carrick, while Nani went on to miss a penalty, but the game was closer than many expected and Terim could not hide his feelings about the result.
"We played well in general but I am upset about the defeat," Terim told reporters.
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"We could have conceded more than one goal. But also, we could have scored an equaliser and beaten Manchester United as well.
"Physically the team did well but we were unlucky in the final touches. Galatasaray have started to play on a different level now and we must develop ourselves.
"I am glad that we fought back against Manchester United but I am upset because we couldn’t reflect our performance in the score."
With Manchester United the early favourites to secure progression out of a group that also includes Braga and CFR Cluj, Terim remains bullish on his side's chances, stating that he is confident of picking up a positive result in the reverse fixture later on in the campaign.
"We wanted to return Turkey with a good result, but my team will continue to work and study this match and learn our lesson," he added.
"Manchester United will be more uncomfortable when we play in Istanbul."