After the match at Griffin Park had ended 3-3 - and 4-4 on aggregate - the hosts won the subsequent penalty shoot-out 5-4 with Miles Storey missing the decisive fourth kick for Swindon.
The Robins had earlier recovered from 3-1 down on the night but were reduced to 10 men in the 101st minute when Byrne was shown a second yellow card for handball.
And the Swindon boss, MacDonald, said: "The character of the team has been fantastic all the way through the season.
"You saw that today but the sending-off turned the game.
"Miles showed a lot of character to step up there. I am proud of him but someone had to miss."
Adam Forshaw converted the winning penalty to send the west Londoners to Wembley to meet Yeovil in the play-off final.