Shortly after City went close again, and beat Bart Griemink in the Swindon goal, but Matt Heywood was able to clear the ball away.
Bristol City scored again just before half time when a Tinnion shot flew in from just outside the box.
The second half was fairly quiet with both sides having few chances, with Swindon never really looking like scoring all game.
Swindon's next game is in the LDV on Tuesday night against third division Southend at the County Ground. Sam Parkin should be back from his 3 game suspension.