Teenage goalkeeper Tyrell Belford was Swindon's hero as Cooper's men defeated their League One rivals 3-1 on spot kicks after a 1-1 draw in normal time at the County Ground.
Belford, 19, playing only his fourth game for the club, saved from Luke Freeman and Filipe Morais to steer Swindon to victory and the Robins will now face Peterborough in the area final with the first leg at London Road in the week commencing February 3.
Cooper hailed his young shot-stopper, saying: "We didn't practise penalties because it's impossible to replicate the atmosphere.
"I picked five players and told them to put their foot through the ball.
"There were some great saves from Ty and we knew which way Morais would go because he has gone there with his last two.
"This is a great chance for us to get to Wembley. I didn't do it as a player, I got nowhere near, but it's something to savour.
"We have a young team and it would be great if we get there because it's something they will remember for the rest of their lives.
"Winning breeds confidence and this keeps our good run of form going so it is a very happy dressing room."