One the face of it, the Black Cats passed up a glorious opportunity to ease their Barclays Premier League relegation fears on Saturday when they squandered a series of chances to claim three precious points against fellow strugglers Crystal Palace.
However, had Kagisho Dikgacoi thumped his late shot just inside, rather than just outside the upright, they would have emerged empty-handed, and a 0-0 draw at least gave them some reward for their efforts.
Head coach Poyet had classified the game as a must-win affair in advance but, having been thwarted, has challenged his players to redress the balance when they head into what represents another six-pointer at Norwich on Saturday.
He said: "That's the way, and now we need to make this point be a good one by going and winning three at Norwich or winning three at Liverpool or wherever it is because, if not, it's going to be like two points dropped.
"I remember at the end of December everybody was saying, 'If you can nick something from Everton and beat Cardiff...'.
"And then it was the other way around and the four points were there somehow, suffering one way and coming back encouraged the other way, because it's like that.
"It's going to be tight, it's going to be a goal, and we need to make sure we don't give that goal away, we don't give any mistakes, any presents to anyone, and we tried our best to do that.
"The future will tell if we are good enough to stay up or not."
Ultimately, another eventful weekend of football saw Sunderland end it one place better off in 18th, but two points further adrift of safety.
Such is the depth of their plight that news of arch-rivals Newcastle's 1-0 defeat at Fulham was greeted by silence rather than the usual roar of approval, with the bottom-of-the-table west Londoners having closed to within a point as a result.
Poyet's men have three games in hand on Fulham, Cardiff and Norwich and two on Palace, West Brom, Swansea, Hull and West Ham, all of whom are within striking distance as a result.
However, they will have to rediscover their touch in front of goal - and quickly - if they are to ease themselves out of trouble.
Three times before the break, Scotland international Steven Fletcher was presented with the kind of opportunities - not gilt-edged, but presentable - which he regularly dispatched without a second thought during the early months of his time on Wearside.
Fletcher did not reappear after the restart, but Poyet later revealed that was due to an injury which represents a significant concern as he reflected upon the striker's misfortune.
The 46-year-old Uruguayan said: "Look, I'm not going to put everything on luck, but yes, sometimes you need an own goal, a deflected shot to go in, whatever.
"It's not going for us, but we will keep trying. The problem is we lost Fletch, I hope with nothing serious, but he twisted his ankle badly and we are going to check on him today and tomorrow to see the extent of the injury.
"But that's why everybody needs to be ready, because the chance is coming for you and you need to take it."
Replacement Jozy Altidore could have won it within six minutes of his arrival, but Palace keeper Julian Speroni pulled off a fine reaction save, although the Argentinian needed the help of the crossbar to repel Fabio Borini's 83rd-minute strike.
It might have been significantly worse for the Black Cats, but Dikgacoi's last-gasp strike flew inches wide to leave Tony Pulis the more satisfied of the two managers.
He said: "Julian had two proper saves to make, so I am delighted with the players. They are very, very tired in there and they have given it everything, and that's all we can ask, all we can ask from them.
"We haven't got the quality that other clubs have got, but the players have got big hearts and they do work very, very hard for me."
- Sports & Recreation
- Barclays Premier League
- Kagisho Dikgacoi