Navarro immediately came back to haunt his former club, after making the switch in the summer, with a second-half double after Paul Benson had given Swindon the lead.
The 31-year-old managed just one goal during 73 appearances at Brighton and Di Canio revealed, in his unique style, that he had told Navarro he expects more this term.
"I told Alan Navarro if he did not score seven or eight goals this season then I would kill him. He has the chance to," Di Canio said.
"He only scored one goal for Brighton over three years before he came here but I know he has the ability to score more goals.
"He gets in good positions and will get chances. He had a bit of luck with a deflection with one of his goals tonight but I was delighted with him.
"I am very happy because we have beaten a very good Championship side. It is a very good win against a good team.
"I was not happy in the first half. Even when I am playing in the garden with my daughter I am not happy if it does not go my way."
Swindon took the lead through Benson eight minutes after half-time, converting Navarro's neat training ground free-kick from six yards.
Navarro doubled their lead 12 minutes later when he slammed home from eight yards after being set up by birthday boy Williams.
Navarro netted again on 75 minutes, hitting a deflected 20-yard strike past former Manchester United goalkeeper Tomasz Kuszczak.
Brighton boss Gus Poyet insisted the only positive from his side's loss was that they could now focus on promotion from the npower Championship, until the FA Cup begins.
"The only positive is that there is not going to be any more cup games," he said.
"We are not going to get distracted by the chance of playing against Premier League teams.
"If we want to play those types of teams then we are going to have to get promoted from the Championship. But there are no positives apart from that.
"They deserved the win and played really well in the second half. People think it is easy when you play games against the lower league teams but it is not.
"Navarro scored just one goal in three years here but it was written in history that he was going to score tonight. We spoke about it before the game.
"We now have to focus on the league. Swindon played really well and deserved the win. They had a good second half and we did not."