The Red Devils had won their five opening Premier League games of the season and took the lead on Saturday thanks to Nani's fine finish on 27 minutes.
However, United's perfect record came to an end as Peter Crouch headed in an equaliser from Matthew Etherington's corner early in the second half.
Fletcher acknowledged after the game that Stoke had performed well but also felt that United had responded strongly to leave with a creditable draw.
"From being in a winning position and losing a goal from a set-piece obviously we're disappointed not to get all three," the midfielder told MUTV.
"It's a difficult place to come. There's always a great atmosphere and you know you're in for a physical match, but I think the lads stood up to it today and battled well.
"You've got to enjoy and relish coming here. If you come here with any fear then that will be evident in your performance. You know what to expect, you prepare for it in training and you try your best to deal with it.
"A lot of work goes into preparing because Stoke do a lot of work on their set-pieces and they're a real threat from them. If you don't do your homework and prepare well you can be caught badly and I think in general we dealt with it quite well.
"Ultimately, I think we maybe gave away too many set-pieces and corners and Peter Crouch is going to take advantage of one of them and that's the only disappointing thing we can learn from."