Bradford, who reached the last eight by beating Wigan on penalties to set up a home tie against Arsenal, had the better opportunities but could not break through a strong visiting defence.
Their only shot on target came in the ninth minute when skipper Gary Jones saw his 20-yard free-kick turned round the post by goalkeeper Richard O'Donnell, although Will Atkinson and substitute Alan Connell saw efforts pass narrowly wide later in the game.
Chesterfield, fresh from their 6-1 home win over Hartlepool in the FA Cup first round, had little to offer in attack until the later stages when substitute Chris Atkinson saw his header saved by the diving Matt Duke before firing just over the crossbar.
