The Black Cats are now five points clear of the drop zone after back-to-back victories as they prepare to face fellow strugglers Aston Villa on Monday evening.
After 11 seasons with Manchester United, O'Shea is more used to fighting at the other end of the table but he says ensuring Sunderland's Premier League survival will rank as a big achievement.
He said: "Success for certain teams can be measured in different ways. If we, Sunderland, survive in the league, it's obviously going to be a success because we were in a precarious position.
"We are still in a small bit of a precarious position, but thankfully we have the power in our own hands to survive that, and we have to continue that.
"It's one of those things, the more games you are winning, the more you enjoy football, so definitely the last few weeks have been a bit more enjoyable.
"It's the same in any sport, whether it be golf or rugby. When you see players with confidence, you can take it to a whole new level.
"The class of the player will always come out on top if they are doing everything properly training-wise, and we have got to stick to that.
"The way we approach games now, we are looking at being a lot more solid, the unit that we have, but also we have been looking at ways to hurt teams, getting more players into the box.
"They are big pressure games we have left, but ones we have been able to cope with and ones we have been enjoying also."