Goals from Alan Connell and Paul Benson secured the win for the Robins and turned the tie on its head at the County Ground, after Callum McManaman had given the visiting Latics a first-half lead.
Di Canio, who can now add this giant-killing to his already colourful career scrapbook, dedicated the success to his late father Ignazio - who passed away towards the end of last year.
And although he was disappointed his dad did not get to witness his latest magical contribution to English football, the Italian insisted that the pair could still bask together in yesterday's glory.
"As I person I want to live in the moment and this is the best moment of my life and career," said Di Canio.
"When you beat a team like Wigan, three leagues above you, in that way from 1-0 down, this is the best moment in my life.
"It matters a lot, not only as a manager but also as a man. The game was live in Italy with everybody watching and my dad couldn't.
"And for this I'm very happy. Even if he couldn't watch it up there (in heaven) I get some satisfaction. My only sadness is that he couldn't watch it with my mum and my brothers.
"I know that he helped me from there and it's a very special day for me."
Di Canio stressed that his team fully merited their 2-1 victory, despite Benson clearly being in an offside position when Matt Ritchie's long-range effort struck the forward's legs and ricocheted into the back of the net.
"When you win you can't win always with a clear shot from 25 yards in the top corner like Ritchie did with Gillingham (in an npower League Two match earlier this season). So you need the bit of luck," he said.
"But why did the luck come? Because Ritchie moved the ball well and had a strike on the goal straight and kept the ball on the ground.
"So obviously the deflection was a little bit of luck but we were there, we were on top and in the second half we were much better than in the first half.
"Many times we had four v four in the box or five v five. Sometimes you need a bit of luck but we didn't win for luck, we won because we deserved to win."
Wigan boss Roberto Martinez rued Lady Luck's contribution to Swindon's victory, but heaped praise on the hosts for the way they set out to play their football.
The Robins more than matched their top-flight guests with their ability to string together intricate passing moves, and Martinez suggested the Wiltshire club have a bright future with Di Canio at the helm.
"From my point of view they had that little bit of luck that you need, the second goal is clearly offside," Martinez said.
"I don't think we had to deal with too much of a threat in the second half but obviously there is a real fighting spirit and a real togetherness there and you can clearly see that there are exciting times ahead for Swindon Town.
"They're not going to be measured by what they do in the FA Cup, they'll be measured by what they do in the league.
"You can see a team that plays with a really high intensity, a real bravery up front. They put bodies forward, they've got real qualities in wide areas - especially with Ritchie and (Paul) Caddis when he comes forward.
"So he (Di Canio) has assembled a very strong side and it's a very exciting time for Swindon."