With chances few and far between in the first half, the Minstermen had Michael Ingham to thank in the 10th minute as he produced a fantastic save from point blank range to deny Shaun Harrad before his defence blocked Russell Penn's goalbound effort to keep the scores level.
Both teams then traded chances with Patrick McLaughlin and Jason Walker going close for the home side, with the Robins only having a tame Jermaine McGlashan effort to their name.
At the start of the second half the visitors' defence needed to be on top form as Walker looked certain to turn in Michael Coulson's centre before Scott Brown kept out Lee Bullock's effort from close range.
With time running out, Cheltenham substitute Kaid Mohamed was inches away from scoring with his first touch but could only fire McGlashan's cutback onto the crossbar before Chris Zebroski headed over late on.
