The Black Cats slipped deeper into the mire on Monday evening when West Ham's visit to the Stadium of Light, which had been identified as a must-win encounter, ended in a costly 2-1 defeat.
Now O'Shea is calling on his team to reproduce their end-of-season surge 12 months ago, which saw them record back-to-back victories over Newcastle and Everton, coupled with draws against Stoke and Southampton, to edge themselves to safety.
The 32-year-old Republic of Ireland international said: "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.
"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."
Such is the depth of Sunderland's plight that two wins from their remaining eight games is unlikely to be enough to save them this time around, and the fixture list has been far from kind.
They launch a run of four intensely difficult away games with a trip to Tottenham on Monday night knowing they could be further adrift of the field by the time they kick off.
O'Shea said: "It's a game once again on a Monday night and there's a little bit of frustration that you are waiting and waiting and waiting for the game, everyone else plays before you.
"But we know what it is and we know if we can put in a performance down there like we have against the teams in the top half of the table that we are going to give ourselves a fighting chance."
Gus Poyet's men have indeed fared rather better against the division's achievers with 17 of their 25 points to date having been taken from sides currently occupying top 10 positions.
However, trips to Spurs, Manchester City, Chelsea and Manchester United suggest their hopes of salvation lie on Wearside, or at least they would do had the Black Cats managed to win more than one of their last nine home league games.
The problem is a basic one: in the six league games Sunderland have played since a 3-0 derby victory at Newcastle on February 1, they have scored three goals and conceded 12, and striking a balance between attack and defence is Poyet's main priority.
O'Shea said: "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.
"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."
Perhaps Poyet's biggest decision will be over whether or not to restore England winger Adam Johnson to his starting line-up having seen him breathe new life into his team and score as a second-half substitute against the Hammers.