The League One club suffered a 3-2 defeat to the Villans at home on Tuesday to bow out of the competition.
"We did our best and I am very proud because we dominated the game," he is quoted as saying by the BBC. "In the first half possession was 60-40 in our advantage, in my opinion, because we played football and they countered only.
"The way we played, I'm more than proud because we dominated not like a typical cup game between a League One side against a Premier League side where the emphasis is on energy and physical presence.
"We dominated by playing football. That makes me very happy as a manager. It was a fantastic achievement for my players."
Di Canio added that he felt that Swindon were the Premier League team, not Villa.
"Talking about the game we looked like the Premier League side and they looked like, with all respect to Aston Villa, the League One side," said the Italian.
"[Christian] Benteke had two clear chances and scored two goals. We had 10-12 and only scored twice. Anyway, we have to accept this result."
Swindon will host Macclesfield Town in the first round of the FA Cup on Saturday.