Lee Barnard, on loan from Southampton, put Oldham ahead in the 68th minute before a spectacular equaliser from substitute Adam Rooney at least prevented a surprise home defeat.
Swindon have not won at home since the middle of January and for long spells they found Oldham hard to break down.
Gareth Roberts should have put Swindon ahead in the 20th minute but hit a post after an Andy Williams shot had been saved by Dean Bouzanis.
Oldham took the lead when Lee Croft crossed for Barnard to shoot into the top corner, but Swindon soon bounced back.
Rooney, who had only been on the pitch five minutes, connected with a perfect overhead kick worthy of his illustrious namesake Wayne to level.