Rovers, bottom of Sky Bet League Two, battled on after Tommy Rowe had been shown a red card just before half-time.
The home side had enjoyed the better of the opening exchanges, with George Broadbent’s low drive forcing Murphy Mahoney into a smart save before the Swindon keeper also denied Zain Westbrooke.
Rowe then fired a shot narrowly over the crossbar after latching onto a loose ball inside the box.
Swindon, though, grew into the game with Charlie Austin and Udoka Godwin-Malife both going close with headers.
The key moment came in first-half stoppage time when Rowe was dismissed for a high challenge on Swindon midfielder George McEachran.
Swindon looked to make the most of their advantage and Austin’s low drive forced a fine save from Rovers keeper Ian Lawlor.
Doncaster, though, remained in the game – and striker Joe Ironside brought an excellent stop from Mahoney as both sides had to settle for a point.