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.
Di Canio revealed after the match that his squad's intense self-belief had exceeded even his own at half-time, when Swindon found themselves level at 1-1.
And the Italian, who branded the victory over Roberto Martinez's Barclays Premier League side as the greatest moment of his career, is hopeful the club's board will act to ensure the win is recognised.
"My lads today deserve something from the club - to do something and put their names in the stadium," he said.
"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. So with this I came into the second half with a feeling we should win the game and it happened."
Di Canio insisted his team were worthy victors despite Wigan complaints that Benson was in an offside position when Matt Ritchie's long-range shot deflected in off the frontman's legs with 14 minutes remaining.
And he was delighted with the reaction of his players after they fell behind just after the half-hour.
"There is no doubt we deserved to win," he said. "Twelve shots on target versus five and six corner kicks versus four, these are only statistics and numbers but they give you an idea.
"We needed to limit their chances to get through and we did that well apart from the accident for the penalty.
"More importantly we were 1-0 down and straight away we pressed forward, we had a chance with Ritchie and we made two or three more good deliveries and then it happened. We deserved to go 1-1 at half-time."
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.
"Today it was pleasing to see two youngsters, Jordan Mustoe and Callum McManaman. I thought they adapted really well and they have progressed really well in the last 18 months and that is pleasing.
"We can't expect to go into the FA Cup using just the 12 or 15 players because that can be very, very damaging.
"We want to be successful, but using the whole squad and today clearly in certain areas we were a little bit slack, a little lacking in minutes."