The Black Cats head for Teesside on Wednesday evening 12 points adrift of safety with only 18 left to play for, and knowing defeat could prove all but fatal to their hopes of avoiding relegation from the Premier League.
However, Moyes, who celebrated his 54th birthday on Tuesday, remains convinced that, despite the odds being stacked heavily against them, they have a genuine a chance of pulling off perhaps their greatest escape yet.
He said: "There's still a chance and while there's a chance, we're going to try to take it. That chance comes at Middlesbrough on Wednesday and we will try to take our chance.
"If we can take our chance there, then we will have another chance against Bournemouth on Saturday.
"We're not stupid, but we are in there fighting and we still think we have got a good chance."
Realistically, Sunderland may have to win all six of their remaining games if they are to stand any chance of avoiding the drop, although Moyes is not allowing himself to look very much further than the 90 minutes at the Riverside Stadium.
With Boro having not won a single league game since December 17 and trailing 17th-placed Hull by nine points with a game fewer left than their north east rivals, defeat for either side could prove hugely damaging.
Moyes admitted: "Look, we don't have enough games left to think that we can let the games slip by. We are well aware of that."
The Scot has found himself in the firing line from sections of Sunderland's supporters in recent months as the club's season has limped towards its conclusion, and even striker Fabio Borini appeared to be making a point when he sprinted towards the bench to celebrate his late equaliser in the 2-2 draw with West Ham last time out in front of, rather than with, his manager.
The former Everton and Manchester United boss said: "I was thrilled about his celebration because he had scored us a goal.
"But I think I'd have really liked him to have gone and picked the ball out of the net and put it in the centre circle because we needed to score a winner to get three points.
"I don't think it's nice when you're not winning most weeks, and that's the thing. You want to win, you come into football to win. At the moment, we are not winning, so that's the hard bit to take."
