Brentford were denied automatic promotion when Marcello Trotta missed a stoppage-time penalty nine days ago and they endured more late heartbreak when Aden Flint's last gasp header forced extra-time at Griffin Park.
The sides remained locked at 3-3, with the tie 4-4 on aggregate, but they were separated in the shootout as Miles Storey's miss allowed Adam Forshaw to score the winning kick, setting up a Wembley clash against Yeovil.
Adam Rooney's own goal and Clayton Donaldson's double had put Brentford into a commanding 3-1 lead, but Joe Devera added to Rooney's goal at the right end before Flint's late header.
"Swindon took us to the wire," Rosler said.
"But the way we played for 120 minutes, just 48 hours after the last match, was magnificent and we deserved to go through.
"After what had happened before, and to concede in the last second again, for them to come back psychologically... they never felt sorry for themselves and they never stopped.
"I am privileged to have a squad like this in my first job in English football."