The Black Cats seemed to be slipping to a damaging defeat when Jonathan Walters put the Potters ahead after only nine minutes and Craig Gardner saw red for a rash lunge on Charlie Adam 11 minutes before half-time.
But their remaining 10 men showed tremendous spirit to outplay their opponents in the second half and got their just reward when John O'Shea's close-range effort secured a 1-1 draw.
The result took them three points above Wigan, who play tonight, and level with Newcastle and Norwich.
Crucially, though, they have a significantly better goal difference than all three and Di Canio believes that leaves them in charge of their own fate.
"We have got destiny in our hands because we are still in a better position than Newcastle and Norwich," he said.
"We have better goal difference so it is still in our hands if we win our games, but I know that is difficult.
"If we lost this game, even more pressure, and obviously a win would have been different - we would have 40 points already."
Speaking after the match, Di Canio described Sunday's home match against Southampton as "a Champions League final" for his side.
But having been impressed by the way his players responded to adversity to salvage a result, he is confident they can rise to such an occasion.
"I saw the fighting spirit I wanted to see and the quality," he said.
"The will is always there but the fact that they were warriors...the shape, the organisation.
"Stoke in the second half had two shots from distance. We dominated the game completely and if you don't have quality you can't do that.
"All these things will be crucial at the end of the season. Sunday will be a difficult game for sure but once again we play at home and if we do the right things we are going to stay up."
Stoke manager Tony Pulis has enjoyed perhaps his most difficult season in terms of wavering support from the terraces but at 11th in the table with two games to go a strong end to the season remains possible.
Some fans have complained about the brand of football employed by the side this season, but Pulis was keen to stand his ground.
Offering a staunch defence of his tenure, he said: "If people want to give their opinion and voice it, then fine.
"But the thing is if you look at the football club over the past six years we've been promoted from being a lower mid-table Championship team, we've been to cup quarter-finals, a semi-final and the final of the FA Cup.
"We've been in Europe and we've had more players than ever, ever before who have played for their countries in the club's 15-year history and we sell out The Britannia every week.
"Expectations have gone through the roof and you have to deal with it.
"There will always be an angle why people aren't happy and you have to accept that.
"If they want to do that I have thick enough skin."