Sunderland slipped deeper into relegation trouble after Monday's 2-1 home defeat against West Ham, which kept the club firmly in the drop-zone, with only eight league games remaining.
Last season the Black Cats found themselves in a similar position, but managed to pull themselves clear with a string of results, including back-to-back victories over Newcastle and Everton.
O'Shea believes the players can repeat this run in the closing weeks, starting with Monday's trip to Tottenham.
"Look, we are not going to give in, we are going to give everything that we can possibly give to make sure we stay a Premier League team," said the defender.
"We got ourselves into a situation last season where we had to produce some big performances and get some big wins, and we were able to do it.
"We hope we are able to do it this time too, it's as simple as that."
With only a handful of games left, O'Shea is aware that Sunderland must go for the jugular against Spurs.
But he insists the club's desperation for points should not force them to adopt a reckless attacking approach on the pitch.
"Look, it's a combination. We want to go forward now, as in we want to get that win from somewhere, and obviously you are giving yourselves a greater chance if you are getting a clean sheet," said O'Shea.
"But ultimately if we don't score, draws are not going to be enough for us. We need wins, so it's that fine balance.
"Yes, we want a clean sheet, but we need the goals as well. As you do, you have to improve and look at yourself and take things on board, that you need to be doing things better.
"But ultimately, it's about sticking together and somehow getting that win from somewhere. Obviously the clean sheet will help it, but ultimately it will be the goals that get you the three points."