Stephane Sessegnon's strike in first-half injury time was enough to secure a second successive victory for Sunderland, who have now eased themselves six points clear of the drop zone.
It was no more than they deserved after creating the better chances in the game with Everton struggling to test keeper Simon Mignolet.
It was Sunderland's first win over Everton in 20 attempts and it lifted them into 14th place and above derby rivals Newcastle on goal difference.
Di Canio said: "I am very happy because it's another win and really important for the squad, for the environment. Two wins in a row - I don't know how long since it happened - to be honest, I didn't check. I don't care.
"It's important that we give consistent performances. That gives you results, gives you points in the table, so that's the most important thing at the moment."
However, Di Canio, who described his emotions during and after his first home game as among the best of his 27-year career in football, refused to concede that one more victory would be enough to secure the club's top-flight status.
He said: "I don't think it's enough. Every weekend, our prediction changes. If Wigan can win against Swansea, they are 34, three points from the main group close to them on 37 with four games to go.
"I don't know, it depends. If you get three points against Aston Villa, it can probably be enough, 40, 41; if you lose against Aston Villa or Stoke and then you beat Southampton, maybe it's not enough.
"I don't know. If we win the direct conflict (with Villa), 40, 41 can be enough, otherwise it's difficult."