Di Canio's side moved three points clear of the bottom three thanks to John O'Shea's equaliser, the point achieved despite Craig Gardner's first-half red card.
Sunderland are level on points with Norwich and Newcastle and have a better goal difference, and their manager has pinpointed Sunday's "Champions League final" against Southampton as key to their survival hopes.
He said: "The game at home to Southampton is now like a Champions League final. It's crucial for our destiny. We can stay up but we need to win another game."
Gardner was sent off for a challenge on Charlie Adam, leaving Sunderland a man short for the second successive game, but the 10 men's reaction was in stark contrast to last week's capitulation at Aston Villa.
Di Canio told Sky Sports: "I saw the team I wanted to see with shape and character. There was only one side in the second half, but we must stop the yellow and red cards.
"We had to keep the shape and use the wide areas better and move Stoke around.
"Also, with our desire and at home we tried to stay in the game and create chances and in the end the desire and effort helped and the point will be crucial to us."
Gardner will not play again this season and Di Canio said: "Craig is out but we can't focus too much on that and we will try not to make any further mistakes. I do love the passion, though, of the players."