They failed to make the most of a series of half-chances which came their way before the break, three of them falling to striker Steven Fletcher, but went close to the opening goal within six minutes of the restart.
Half-time substitute Jozy Altidore initially miscontrolled Wes Brown's long ball, but having regained possession, span and fired in a shot which Palace goalkeeper Julian Speroni had to tip over.
However, it was Fabio Borini who came closest to breaking he deadlock as time ran down, rattling the crossbar with a stinging 83rd-minute shot and then only just missing the target when the ball fell to him once again.
Kagisho Dikgacoi might even have won it for the visitors at the death, but blasted his effort inches wide to spare the Black Cats further punishment.
Asked for his reaction, Poyet, who revealed Fletcher had been withdrawn because of a twisted ankle which will be assessed over the next 48 hours, said: "A word you will probably never hear again from me, very calm. I feel calm today.
"We tried everything. That's what we are, that's the team that we have, the players we have got and the way that we need to play.
"If we play like that, the future will tell if we are good enough to stay in the Barclays Premier League or not.
"But that's the way to perform, to take the game to the opposition, to assume risk, to be in control, to try crosses, free-kicks, shots, in behind, in front, one-twos - we tried everything."