Trailing 1-0 from the first leg Swindon took 81 minutes to get back into the tie. Sam Parkin grabbing the vital goal that sent the match into extra time.
Swindon dominated the first period of extra time and took a deserved lead through Rory Fallon. But with only seconds left in the tie Brighton fullback Adam Virgo headed the ball past Swindon keeper Rhys Evans to level the scores on aggregate and send the match to penalties.
Former Bristol Rovers player Andy Gurney missed the deciding spot kick as he blasted the ball against the post.