Referee Darren Drysdale took centre stage by showing Elliott a straight red card following an off the ball clash with Swindon’s Southampton loanee Jack Stephens.
Swindon took until the 78th minute to make their man advantage tell when Michael Smith showed great footwork inside the penalty area before picking out the bottom right corner.
Smith’s goal was a rare moment of quality in a scrappy game between the sides third and first in the League One table.
Nathan Byrne had come within inches of giving Swindon the led when his 25-yard strike rebounded off the crossbar.
City goalkeeper Frank Fielding also produced an excellent save to turn Stephen’s low shot behind for a corner.
The league leaders had few clear cut opportunities of their own and Joe Bryan’s 25-yard effort into the side netting on 49 minutes was as close as they came to scoring.