Bridges opened the scoring for Carlisle after only 8 minutes with a goal from just inside the area.
The second half saw Swindon looking a much better side with Jack Smith coming close for Swindon after hitting the post.
Swindon left it very late to equalise, but 3 1/2 minutes into the 4 minutes of stoppage time Swindon were awarded a free kick just outside the area. Tudor-Jones took the kick, which was then slightly deflected and scored to make it 1-1.
The draw which was well deserved leaves Swindon in 17th place on 49 points with just 2 games to go, however most sides around them still have games in hand.