The Pilgrims led through Brad McGowan on the half-hour mark but were eventually undone by Bradford replies either side of half-time from Mark Ross and Adam Boyes.
It sentenced United to a fifth reversal in six league away days during the opening weeks of the new campaign.
Monday`s other game saw Leamington pick up another encouraging point from a tough away game.
Just 48 hours after their battling display in a goalless draw at Darlington FC, the Brakes shared a 1-1 draw at the Tameside Stadium against Curzon Ashton.
Leamington found themselves behind on 28 minutes when Ben Wharton spread the play on the right to Max Leonard, who cut into the box and saw his cross blocked by a sliding challenge from Jamie Hood, the ball unfortunately sitting up perfectly for Wharton to smash between keeper Tony Breeden and his near post.
The hosts saw a couple of long-range efforts fly wide of the target early in the second half before Leamington began to grow into the game once more.
Their persistence paid off with 19 minutes remaining when Jack Edwards (pictured) connected with Coby Bishop’s knock down from a cross into the box to slam the ball into the net off the inside of the post.
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