It was Cheltenham who showed up first when Paul Connor burst through the middle to shoot wide of the advancing Petr Brezovan.
A howling gale and heavy rain did nothing to improve the quality of the football but despite all the off-field problems Cheltenham gave a very spirited display.
There were few real chances early on but in the 32nd minute Hal Robson-Kanu robbed a defender in midfield and went on to hit a low left-foot drive passed Cheltenham keeper Scott Brown.
Cheltenham responded immediately when Josh Low fed Elvis Hammond, who went on to smash an 18-yard drive into the top of the net.
Cheltenham's season got significantly worse just before half-time when the talented Ashley Vincent, who was looking good for the visitors, was stretchered off following a challenge by Michael Timlin, who was booked.
It transpired that Vincent had damaged knee ligaments and Martin Allen, the Cheltenham boss, will await a fuller report in the morning but he described it after the game as a "bad one''.
Cheltenham showed plenty of purpose after the interval and when Swindon failed to clear Dave Bird's long throw Hammond snapped up the chance from six yards in the 54th minute.
Town boss Danny Wilson made a tactical substitution replacing Jamie Vincent with Kevin Amankwaah.
Amankwaah immediately made his presence felt when he headed powerfully past Brown from Robson-Kanu's corner just two minutes after coming on in the 66th minute.
Both sides might have pinched a win with Swindon's Jack Smith forcing an excellent save by Brown and at the other end Brezovan came to Swindon's rescue with a timely block from substitute Nicholas Bignall's effort.
Cheltenham held on in the final minutes for an important point but both sides will need to do considerably better if they are to avoid the drop to League Two.