A Burak Yilmaz header settled the match in the second half but a young Red Devils team had their share of the chances.
United went into the game with top spot in Group H already tied up, meaning that Sir Alex Ferguson was able to rest key players, but Fletcher still believes that the visitors warranted at least a point.
Speaking to Sky Sports after the match, the midfielder argued: "There were difficult conditions. The pitch wasn't great, it was an intimidating atmosphere.
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"We came here and I think we played extremely well, especially in the first half. As soon as they got the goal they slowed the game down, committed fouls, a lot of tactical fouls.
"They got behind the ball in the second half, which made it difficult, but we really should have got the goal in the first half as we had great opportunities on the counterattack."
Phil Jones, who made his first appearance of the season in Turkey, was in full agreement with his team-mate, underlining the fact that United had enough chances to score.
"We should have switched on and dealt with the ball but it does happen," he added of Burak's goal. "We got punished from a set-piece and that shouldn't happen, it's not our game.
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"We need to have a look at the video and see where we went wrong. I thought we played well tonight in patches and thought we were unfortunate not to take the lead.""I'm glad [to be back]," the returning defender continued. "I've had a bit of a torrid six months. In and out of the team, can't get fit, bad injuries and bad luck but I'm delighted to get 90 minutes under my belt and hopefully I can push on now and prove why I should be here."
Much of the pre-match talk centred around the notorious atmosphere generated at Galatasaray matches but Fletcher highlighted the poor quality of the pitch as the biggest problem.
"It was difficult [communicating]. Shouts like 'man on' and 'turn' were a bit difficult to hear but you've got to experience that, you've got to deal with that," the Scot remarked.
"There were problems with the pitch. The pitch wasn't great at all. It was difficult to play one-twos and quick passing because the ball was bobbling up but I think, especially in the first half, we did enough.
"All the young players who played tonight I think did fantastic and that will stand them in good stead for the future."