In a typical cup tie which was battered by blustery winds, neither side created many clearcut chances, meaning they will go head-to-head again in a first round replay at Valley Parade.
The National League outfit were on the front foot for much of the first period as City struggled to cope with the conditions.
Richard Brodie was thwarted in the box inside the first two minutes, before a fantastic challenge from Reece Burke denied Charlie Walker a one-on-one with the goalkeeper.
Jim Stevenson then hit two decent efforts from distance, the first firing narrowly wide after a deflection, and the second sweetly struck half-volley saved well by Ben Williams.
The visitors’ best chance came two minutes before half-time as the ball fell to Billy Knott in the area, only for Omar Beckles to head his shot over the bar.
Bradford bossed the game after the break, starting the second half in a much more spritely style as Billy Clarke turned his man before firing in a decent effort from a tight angle which Phil Smith was equal to.
Aldershot midfielder Damon Lathrope almost got lucky with the wind as his free-kick threatened to drop under the bar, but Williams gratefully watched it just over.
Shortly after coming on as a second-half substitute, Devante Cole spun and shot inside the box, only to see his strike stick to Smith’s gloves, and City continued to be stifled by their spirited opponents for the final 15 minutes.
Aldershot: Smith, Alexander, Beckles, Saville, McGinty, Walker, Stevenson, Lathrope, Browne (Richards 73), Walker, Brodie. Unused substitutes: Oliver, Hatton, Thomas, Barker, Mavila.
Bradford City: Williams, Darby, McArdle, Burke, Meredith, McMahon, Knott, Liddle, Reid (Marshall 84), Hanson, B. Clarke (Cole 70). Unused substitutes: N. Clarke, Leigh, Mottley Henry, Cracknell, James.
Referee: Ben Toner.