Goals from Alan Connell and Paul Benson earned a 2-1 win for the npower League Two outfit after Callum McManaman had handed the visiting Latics a 34th-minute lead.
"My lads today deserve something from the club - to do something and put their names in the stadium," said Di Canio. "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."
He added: "(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. So with this I came into the second half with a feeling we should win the game and it happened."
Wigan boss Martinez praised Swindon in the aftermath of an embarrassing defeat for the Latics, who started the afternoon 53 places above their hosts in the Football League pyramid.
"I thought we started well and we got a well-deserved lead but I thought Swindon reacted really well to the emotions of the crowd," he said.
"They got a little bit of luck with the second goal which looked offside but it was a great opportunity for me to see the whole squad and all my players.
"Swindon have a real togetherness, have a real fighting spirit and they showed what it means to be in the FA Cup.
"I'd never say that we're not interested in the FA Cup, we want to win every game that we go into, but it's true that to be in two competitions you need to use your whole squad."
Source: PA
Source: PA