Nelson sent the ball to Phil Smith's right in the Swindon goal to put his team through.
The result was heartbreaking for Swindon, who had fought back from 2-0 down after 25 minutes to draw level by the 70th minute. But neither team could break the deadlock in extra time and it was left to a penalty shoot out to see who would go through to the next round.
Brentford skipper Kevin O'Connor opened the scoring on eight minutes. Paul Brooker put over a corner from the right hand side, former Swindon defender Matt Hayward flicked on and O'Connor ghosted in at the far post to bury the ball past Smith in the Swindon goal.
Brentford increased their lead on 24 minutes with O'Connor turning provider. Brooker sent him down the right hand side and his pinpoint cross was hit first time by former Arsenal trainee Jo Kuffour into the bottom right-hand corner of the net.
Swindon hit back before the interval with an Andy Nicholas goal as he headed in Aaron Brown's corner from point-blank range.
Brown and Paul Evans both went close for Swindon in the second half as the hosts pressed for the equaliser which eventually arrived on 70 minutes. Evans picked up a loose ball in midfield, ran forward and planted a 25-yard left-foot shot into the left-hand top corner of the Brentford goal.
Swindon continued to look for the winning goal but found it hard work breaking down a resolute Brentford defence.
Brentford almost won the game in injury time when Kuffour's right-foot shot was well saved by Smith, diving down to his left.
Evans and Royce Brownlie both went close for Swindon in the first half of extra time.
In the second Fola Onibuje's cheeky back-heel almost won the game for Swindon, but Nelson scrambled back to make the save.
Swindon striker Brownlie was then sent off 45 seconds before the end of the match after he was shown a second yellow for dissent.