The visitors scored early on and then had to withstand almost constant pressure as Chester toiled in vain for an equaliser, despite enjoying the majority of possession.
The game had got off to a flyer with the visitors taking the lead within 90 seconds of the start, only moments after Chester had an opportunity to go in front.
Ryan Lowe had an effort cleared off the line by Craig Armstrong, before Cheltenham broke quickly down the other end of the field and Brian Wilson slotted the ball past goalkeeper Paul Harrison for the opener.
Despite falling behind, Chester had looked the brighter of the two sides and on 21 minutes they almost carved out an equaliser as Lowe was only inches away from getting on the end of a great cross from Ben Davies.
Five minutes from the interval captain Stewart Drummond hit a great volley on the turn which was well saved by Shane Higgs.
Chester had a good opportunity two minutes after the re-start when Davies crossed to Lowe who didn't connect properly with his attempted shot, allowing Higgs an easy save.
It was the home side that continued to make most of the running without seriously threatening the visitors' goal.
Chester manager Mark Wright opted to shake things up by throwing substitute Marcus Richardson into the fray in place of Gregg Blundell just after the hour mark, and the tall striker started to pose a few problems with his aerial threat.
On 74 minutes Michael Branch lost his marker and then completely missed the ball with an attempted header at the near post, and then the forward shot wide when only 5 yards out as the best chance to salvage a point went begging.