Sunderland took the lead in the first half through Stephane Sessegnon, and, after a Papiss Cisse equalizer was incorrectly ruled offside, the Black Cats further strengthened their advantage with exceptional strikes from Adam Johnson and David Vaughan.
Di Canio was full of praise for his side after the game, and he believes his players showed the character needed to stay in the Premier League.
"We had to not be shy, not to only have two lines very close together and not challenge in the 50-50," Di Canion told reporters. "We are here to fight for our future and we did that in the end.
"My players did a fantastic job, they were warriors so they deserve this fantastic result."
The victory moved Sunderland to 15th place, three points clear of the bottom three, but the Italian admitted the win - despite being against the club's bitter rival - would mean little at the end of the season if his side is relegated.
"I won many important games [with Swindon] in the cup, so that was an important moment for the club's history," Di Canio said. "But when you arrive at the top in the Premier League and you win one of the toughest derbies in the world...to do this after many years...we won 3-0, [we scored] three goals, [and kept a] clean sheet. Obviously as a single game [it] is the best and the most important, but it's nothing if we don't stay up."
Di Canio was typically animated in the Sunderland technical area, sliding to his knees in celebration and enjoying the plaudits of the travelling support after the game, but insisted most of the credit must go to his players.
"I wanted to go far, but I couldn't because I am too old," Di Canio joked. "I only have 20 yards in my legs nowadays. There was a good moment with the fans, we were really happy.
"They [the fans] have to sing the players' name because the players worked so hard. I shuttle 10 yards up and down the technical area because I have energy to do this.
"It's more important for my players because they give everything. They were heroes, they did a fantastic job."