The Addicks' captain Nicky Bailey blazed wide from 12-yards in what proved the only missed spot-kick of the shootout, with Town converting all five of their pressure kicks to set up a Wembley showdown with either Millwall or Huddersfield on Saturday May 29.
GALLERY:Photos from The Valley
Leading 2-1 from the first leg at the County Ground last Friday, the FourFourTwo-sponsored side's luck deserted them early on when goalkeeper David Lucas had to be replaced after falling awkwardly and injuring his shoulder.
Danny Wilson's outfit then saw their slender advantage wiped out by the break, after an own goal from Simon Ferry and fine finish from David Mooney fired the hosts into a 2-0 lead on the night - 3-2 on aggregate.
Swindon appeared dead and buried midway through the second period as captain Gordon Greer saw red for a high challenge on Deon Burton.
However, against the odds the Robins rallied and levelled the tie through Danny Ward - racing onto a loose ball and prodding it past goalkeeper Darren Randolph.
Miguel Llera then received his marching orders for Phil Parkinson's side in injury-time after dragging down Charlie Austin as the prolific former Poole Town forward surged towards goal.
Neither side could be seperated after an additional 30 minutes extra-time, although it was the Addicks who posed the greater threat with stand-in goalkeeper Phil Smith and the woodwork keeping Town in the tie.
And it was the visitors who proved the cooler penalty takers, as Jon-Paul McGovern, Charlie Austin, Kevin Amankwaah, Danny Ward and Stephen Darby all scored to send the 3,000-strong travelling support into delirium.
WIN:A playing contract with Swindon
Afterwards, Wilson told Sky Sports: "That was a real, real gritty performance from the boys. After the sending off they showed their character, like they have done all season, and to get a goal in that period as well was a bonus for us.
"Overall I thought it was nip and tuck, we had a bit of the play and then they did. There were two good sides very committed out there and unfortunately going out on penalties is not always fair but you have to get a result somehow and thankfully we were the recipients of it tonight.
"I think it was in the stars, it was down to ourselves. We said at half time don't even think about getting two goals, just get one and see what happens.
"The whole atmosphere changes when you score, like we saw today, and after being in the ascendency for long periods Charlton took a bit to get back after conceding a goal. It gave us a lift and we found an extra yard and a bit more energy."
