The 18-year-old has made nine Town appearances and scored three goals in his last two.
Di Canio told BBC Wiltshire: "Everybody thinks he's ready for the top level. No. One day, probably, he'll be ready but he'll have to work very hard.
"He's at the base of his improvement."
Storey, who scored his first professional goal in Swindon's
4-0 win at Stevenage on Saturday
, struck twice in three minutes to set up a possible fourth round upset against Villa, before Christian Benteke hit a late winner for the Premier League side.
But despite the young striker's recent good form, Di Canio says a regular place in the side is far from guaranteed.
"I have to use him in a way that I think I can use him," the Italian said. "Next game maybe he'll be on the bench or play half an hour or 20 minutes.
"Otherwise we'll lose the sense of reality. We'll only damage him if we think now he can play League One."
Storey broke into the Town side at the start of last season, but after three successive appearances in September was made to wait until the final day of the campaign for another opportunity.
"Last year it was hard when I brought him up (to the first team)," Di Canio said. "After two months I sent him back to the youth team because he didn't have this education as he was young as a footballer.
"In the summer we were thought with a full emersion with us and the rest of the professional players, he will improve and maybe understand more."
Following his brace against Villa, Storey has been placed on standby for England Under19s for their forthcoming international match against Finland.
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