Goals from Alan Connell and Paul Benson gave the League Two side a famous victory after they had gone a goal down, and Di Canio wants the board to act upon what he has hailed as the greatest moment of his career.
"My lads today deserve something from the club - to do something and put their names in the stadium," said the manager.
"Normally you do this when you win something important, but I don't want a big statue. Maybe just a small plaque with all their names because they did something special.
"(At half time) I said to them once we go forward, attack because otherwise it is difficult to get what you want today, let the dream come true, and today they did show the dream can come true. But what was a surprise to me was that at half time they believed more than me."
Swindon's win means the club will appear in the FA Cup fourth round for the first time since 1996.