The Robins have spent much of the season to date in the bottom four but showed that they have the stomach for a battle that might just lift them clear of trouble.
Luton, on the other hand, saw plenty of the ball but created little with it against determined opponents far more used to life in the League One danger-zone.
Luton's fortunes may have improved with their FA Cup run but they will need to match the workmanlike efforts of sides like Cheltenham if they are to overturn the ten-point penalty handed them for going into administration.
Cheltenham made the perfect start with a goal after only two minutes. David Bird's long throw from the left was flicked on by Shane Duff and midfielder Andy Lindegaard stole in front of his marker to flick home a shot at the near post.
Luton hit back immediately and Cheltenham goalkeeper Shane Higgs was forced to make saves from Chris Coyne and Matthew Spring but the home team should have added a second goal midway through the first half.
Damian Spencer's right-footed blast was blocked by the legs of goalkeeper Dean Brill. The ball spun up into the air and, with the goal gaping, Cheltenham strikers Steven Gillespie and Guy Madjo somehow managed to get in each other's way and the chance was lost.
Higgs saved again from Spring towards the end of a frantic first half in which neither side was able to play any fluent football.
The second half quickly settled into a pattern of Luton pressing forward, although Cheltenham almost scored on the breakaway when Gillespie sprinted from deep inside his own half only to see his shot blocked by Brill.
The home side stood firm against a barrage of Luton pressure with central defender Shane Duff holding them together at the back, although they had Higgs to thank again deep into stoppage time when he kept a header from Calvin Andrew out with a fingertip save.