The League Two high-flyers enjoyed the better of the chances and could have caused an upset as they became the only second club this season to prevent Leeds from scoring a goal.
But Hereford's famous FA Cup fighting tradition lives on and The Bulls will now go to Elland Road for a potentially lucrative replay, but must feel that they could have snatched the tie out of the fire in the closing stages.
The lively Lionel Ainsworth forced a brilliant save from goalkeeper Casper Ankergren with a leaping header from Clint Easton's cross.
And then centre-back John McCombe's hooked volley drifted agonisingly wide of the upright with Ankergren beaten. Leeds had gone on the attack from the kick off and Jonathan Howson's early shot tested Wayne Brown.
But it was the Bulls who had the better of the early exchanges with the pace of Ainsworth and Theo Robinson stretching the Leeds rearguard.
The visitors, however, improved as the first half progressed and a shot from Jermaine Beckford narrowly cleared the bar, before Tresor Kandol scuffed his shot straight at Brown from a dangerous position.
At the start of the second period, Hereford twice came within inches of breaking the deadlock inside the first minute of the restart.
A cross from Easton was sliced over his own bar by Paul Huntington and, from the resulting corner, Trevor Benjamin's header clipped the outside of the post.
But Leeds continued to threaten and a fierce 20-yard drive from Ben Parker forced an excellent diving save from Brown.
Hereford suffered a blow just after the hour mark when skipper Karl Broadhurst was stretchered off with an ankle injury after a challenge on Beckford and now faces a month on the sidelines.
But Hereford were not thrown out of their stride and it was certainly the Bulls who finished the game more strongly.