Tommy Rowe scored the crucial spot-kick as Posh sealed a trip to Wembley after Massimo Luongo and Tijane Reis had blazed over for Swindon.
Delighted Ferguson, who won the competition as a player with Wrexham, said: "It is a big win for us.
"Getting to Wembley captures everyone's imagination so it was a big night for the town and for the club.
"We worked on penalties in training, we spoke about it and I said to the players if you are in any doubt just make sure you get a good contact with the ball.
"They were good quality penalties and on the night I felt we deserved the win.
"We pressed them, especially in the second half but it has taken a great finish from Britt (Assombalonga) to rescue us.
"We are delighted. I won the tournament as a player and it was one of our targets at the start of the season.
"Once you get to Wembley the desire to get back there in the play-offs will be massive."
Swindon were denied a penalty as Craig Alcock's fail on Dany N'Guessan went unpunished but the Robins took the lead 13 minutes before half-time after some comical Posh defending.
Luongo whipped in a teasing left wing cross and Alcock smashed a clearance against N'Guessan before Alex Pritchard tucked home the loose ball.
N'Guessan hit a post before Assombalonga drew Posh level 15 minutes from time with his 24th goal of the season, coolly heading Michael Bostwick's cross beyond Wes Foderingham.
Luongo set the tone for a miserable shootout for Swindon by smashing the opening spot-kick into the stands.
Foderingham gave Mark Cooper's side hope by saving Danny Swanson's effort only for substitute Reis to also miss the target.
Swindon boss Cooper said: "I thought we played well and did enough to win the game, we made one mistake and it led to a goal.
"My players gave everything and were deservedly in front at half-time but it was not to be. If you give Assombalonga one chance he will score.
"We played well and fought and you have to hope luck goes you way but it didn't.
"We have got a lot of disappointed young men in the changing room and it my job to get them up for a tough game at Leyton Orient on Saturday."
Cooper also revealed striker Nile Ranger will be out for up to 12 weeks with a hamstring injury.