Sam Parkin had sent the game into extra time when he scored on 81 minutes and Rory Fallon's goal appeared to send Swindon en route to a Cardiff meeting with Bristol City.
But Adam Virgo scored in injury time of extra time to send the contest to a shoot-out and he also netted the decisive spot kick as Tommy Mooney and Andy Gurney missed for Swindon.
The first half was a tense affair with the wet conditions hardly helping both sides to settle their nerves.
Charlie Oatway almost conceded an own goal when diverting a Swindon free kick into the side-netting after goalkeeper Ben Roberts flapped at the cross.
The conditions also played a part in Roberts failing to claim Matt Hewlett's through ball under pressure from Parkin but there were sufficient Brighton defenders in attendance to clear the danger.
Brighton soaked up the bulk of Swindon pressure but survived another scare towards half time when Virgo cleared Mooney's attempt off the line.
Roberts again misjudged a visiting free kick early in the second half while Swindon had the hosts' prolific striker Leon Knight firmly under lock and key.
When Knight was forced off midway through the second period, his replacement Adam Hinshelwood almost scored a decisive goal as he drew a fine point-blank save from Rhys Evans.
The Seagulls' decision not to press for a killer goal almost cost them on 78 minutes when Mooney's searing drive clipped the top of the crossbar.
Just three minutes later and Brighton were made to pay, largely thanks to the industry of Sammy Igoe.
When Igoe's shot was parried by Roberts, he followed up to lay the ball across the six-yard box and, after a brief scramble, Parkin scored from close range.
The fraught contest edged into extra time and it was Swindon who proved the stronger after capitalising on Brighton hesitancy.
Mooney picked up the ball just inside the visiting half before surging towards the penalty area where he slipped in Fallon to clip the ball beyond Roberts.
Brighton fans then mistakenly believed their team had pulled a goal back but John Piercy's shot brushed the wrong side of the netting.
The Robins could have wrapped up the tie before the end of the first half of extra time with the influential Igoe picking out Mooney but his shot was smartly pushed out by Roberts.
The hosts were unable to inject verve into their game following their earlier cautious approach although Chris Iwelumo could have done better than to shoot straight at Evans and it was a similar story for Hinshelwood who headed over from six yards.
But in a dramatic conclusion, Brighton took the match to a penalty shoot-out in injury time of extra time when Danny Cullip flicked on a long throw-in and Virgo stole in to head past Evans as the prelude to a thrilling win for birthday boy Mark McGhee's side.
