James Hanson, Gavin Grant and Michael Flynn missed gilt-edged openings before Frenchman Mathieu Manset put Hereford in front in the 33rd minute.
Hereford goalkeeper Adam Bartlett also made some good saves to deny the Bantams before teenager Jake Jervis scored the Bulls' second goal when nine minutes remaining.
"It's been frustrating for me. I am not saying we deserved to win but we did create more chances," said Taylor.
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"Their defenders made a couple of good blocks but we had some free headers and I thought we had the better chances. But overall in open play I didn't think we played well enough after a quarter-of-an hour, which was okay. After that, it was not good enough."
Bradford had a chance to go in front in the early stages but Hanson headed wide of the far post and then Bartlett palmed away a rising shot from Grant.
Flynn also fired over the top before Manset headed Hereford in front from close range.
After the break, Gareth Evans had a gilt-edged chance to level but he shot over the bar.
Striker Hanson also had a powerful drive palmed away before substitute Mark McCammon headed against the post.
Hereford struck again nine minutes from time when Jervis lobbed goalkeeper Matt Glennon to end the Bulls run of four consecutive home defeats.
Hereford United: Bartlett; Green, Valentine, Rose, Lunt, Pugh, Young, McQilkin, Jones, Manset (Constantine 71), McCallum (Jervis 71). Unused substitutes: Sonko, Gwynne, Preston, Done, Adamson.
Bradford City: Glennon; Flynn, Clarke, Bullock (McCammon 59), Evans (Kendall 77), Bateson, Hanson, Threlfall, Grant (Daley 51), Oliver, Bolder. Unused substitutes: O'Brien, Rehman, McLoughlin, O'Leary.
Referee: D McDermott (London).
Man of the match: Mathieu Manset.
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