Two goals in two second-half minutes in a game dominated by the gale-force conditions were enough for Neil Warnock's side to extend their unbeaten run to four matches.
For an hour City battled gamely as they looked to extend their own three-match unbeaten streak, and they could perhaps have considered themselves favourites for at least another point after overcoming the howling gales during the first half.
Those winds were almost certainly replaced by a blast of hot air from Warnock during the break after hearing his side booed off the pitch when the half-time whistle sounded.
Although Dean Windass won arguably an important coin toss and chose to kick into the wind, Warnock was within his rights to expect an onslaught.
But it failed to materialise, with United failing to put goalkeeper Clint Davies under any pressure as the made his full debut.
Robson had been forced to turn to Davies with Alan Combe suspended, Mark Paston injured and Nico Vaesen now with Crystal Palace.
Davies made just one save, and that did not arrive until nine minutes from the interval with Michael Tonge firing a 20-yard drive into the rookie's midriff.
Tonge had earlier been given a free-kick opportunity in the 12th minute after Peter Atherton had been booked for bringing down Jack Lester, but his attempt was ballooned into The Kop.
Remarkably, it was the Bantams who came closest to forcing the opener as they battled gamely in the face of the gale, with Wales international Paul Evans almost making Warnock grimace.
Evans first came close in the 20th minute with a left-foot curler which nearly deceived Kenny, while four minutes from the break he thundered a volley from 20 yards inches past Kenny's right-hand post.
The jeers which greeted the United players as they trudged off should have acted as a spur, but for the opening 15 minutes the home side continued to struggle as they tried to overcome the conditions.
Then came a vital double substitution as Warnock brought off Nick Montgomery and Dries Boussatta, replacing them with Izale McLeod and Jonathan Forte, and it was the latter's surging run within seconds of his introduction which led to him winning a vital corner.
Tonge played a short ball to Stuart McCall, and his return was swung in by the left winger for Chris Morgan to plant a stooping header beyond Davies for his first league goal since his free-transfer summer move from Barnsley.
Two minutes later and the points were sealed with Lester capitalising on a mistake by Mark Bower to curl beyond an advancing Davies his 14th goal in 28 appearances this season.