Ashley Ward scored his first goal in City colours as the reserves went down at Bury's Gigg Lane ground in their first match of the new season.
City lost a 2-1 lead in the second half after fighting back from 1-0 down.
United's winner came 12 minutes from the end, scored by Swedish Under-21 international Bojan Djordjic.
Ward scored City's first equaliser with a fine far-post header from Peter Beagrie's cross after 29 minutes.
The £1.5million signing from Blackburn, who had earlier gone close with a shot that whistled inches wide of the post, also impressed with his mobility and smart passing before he was substituted at half-time with tomorrow's Premiership match against Leicester City in mind.
Beagrie also went off after 57 minutes for the same reason.
City gave a debut to Italian midfield trialist Sergio Campolo, who is a friend of Benito Carbone. He is a free agent after playing for Perugia in Serie A.
Campolo linked up well in midfield in what proved to be an enjoyable match with both sides committed to the passing game.
The entertainment on show augurs well for the new season in the FA Premier Reserve League which offers an attractive fixture list comprising the top clubs in the north.
United fielded a predominantly young side with out-of-favour goalkeeper Mark Bosnich the only senior player.
They went ahead after 24 minutes following a free kick from midfield player Jonathan Greening who was linked with a move to City during the summer.
The ball found Alex Notman who appeared to have taken the ball too wide, but he managed to score from a narrow angle.
The sides were level 1-1 at half time following Ward's equaliser.
City went in front after an hour with a superb header from substitute Ben Jones who rose above the United defence to meet left back Wayne Jacobs' cross.
But Notman equalised with a far-post header from Danny Webber's cross seven minutes later.
City then paid the price for failing to cut out another cross to the far post after 78 minutes, this time from David Healy, and unmarked Djordjic fired home the winner.
Former United goalkepeer Gary Walsh kept City in the game in the first half with superb saves to deny Healy and Norman.
He proved he is back to somewhere near his best form after missing the second half of last season following two knee operations.
Walsh also prevented a certain goal five minutes after the interval with a brilliant reflex save to keep out Alan Tate's downward header after City's defence had failed to deal with Djordjic's corner.
Manchester United: Bosnich, Lynch, Studley, Clegg, Tate, Stewart, Davis, Greening (Webber 66mins), Healy, Notman, Djordjic. Subs (not used): Cosgrove, Rachubka, Evans, Clegg.
Bradford City: Walsh, Halle, Jacobs, Kerr, Myers, Bower, Lawrence, Campolo, Ward (Jones 45mins), Grant, Beagrie (57mins). Subs (not used): Davison (gk), Hardy, Fishlock.
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