Neither keeper was overly worked throughout the game, but it was Sunderland who came closest to snatching a winner through Fabio Borini.
The Italian rattled the crossbar with seven minutes remaining in a game which had earlier seen Palace keeper Julian Speroni make a vital 51st minute save from substitute Jozy Altidore.
Victory would have taken Sunderland above the Eagles on goal difference, but the home side passed up a series of chances to make their dominance tell with Steven Fletcher particularly guilty before he was replaced at the break.
The Black Cats have won just one of their last eight home league games, and might have left-empty-handed had Kagisho Dikgacoi not wasted a glorious late chance to win it for the visitors.
It was a game Sunderland really had to win if they are to launch a successful fight against the drop, and they made a fist of it in the opening 45 minutes as they took the game to Palace without ever looking like conceding.
However, they rarely looked like scoring either as the problem which has hampered them for much of the campaign to date showed little sign of abating.
Poyet has been searching since his arrival in October for a man to spearhead his attack with either Fletcher or Altidore usually employed in the lone striking role which was handed, with some success, to Fabio Borini in the Capital One Cup final.
It was Scotland international Fletcher who got the nod once again on Saturday, but opportunities - admittedly half-chances - came and went.
The former Wolves frontman failed to connect properly with Adam Johnson's first-minute cross and then under-achieved with two audacious back-heeled attempts from deliveries from Borini and Johnson.
They were the kind of chances he dispatched with aplomb during his early months on Wearside, but it is simply not happening for him in front of goal when his team needs it most.
Full-back Marcos Alonso had earlier thumped a long-range effort into Speroni's midriff and Borini volleyed acrobatically wide from Thomas Ince's miscued clearance two minutes before the break.
For their part, Palace, who had full-backs Adrian Mariappa and Joel Ward both booked inside the opening 13 minutes, offered little going forward, and their only real opening came courtesy of a slip by Vito Mannone.
The Italian stumbled as he attempted to clear and bundled the ball to Ince, but he recovered in time to claim the midfielder's tame 31st-minute shot.
Fletcher was put out of his misery at the break when he was replaced by Altidore, who might have made his mark within six minutes.
The United States international miscontrolled Wes Brown's ball over the top, which looked to have put him in on goal, but recovered his composure to spin and fire in a left-foot shot which Speroni managed to tip over.
Borini volleyed high over from Liam Bridcutt's cross four minutes later with Sunderland continuing to dominate, although they were served warning of the visitors' threat on the break within seconds when Yannick Bolasie forced a save from Mannone.
However, there was little or no quality from either side in the final third as Speroni and Mannone were virtual spectators.
Cameron Jerome might have caught out Mannone with a clever near-post flick from Ward's 75th-minute corner, but the Italian was well placed to pluck the ball from underneath his crossbar.
Borini came closest to breaking the deadlock with seven minutes remaining when he ran on to Ki Sung-yueng's through-ball only to crash a rising drive against the bar with his follow-up flying inches wide.
Palace's best effort came in the last few minutes when Jerome played in Dikgacoi on the edge of the six-yard box but he blazed inches wide when he could and should have hit the target.