The Eagles came closest to winning the contest through Luka Milivojevic's first-half free kick which hit the post, while Mamadou Sakho also squandered a glorious opportunity towards the end of a scrappy affair.
United are still winless in the league this term and have failed to pick up all three points in their last six away outings stretching back into last season, meaning that they remain inside the division's bottom three.
Palace were almost ahead inside the opening 60 seconds, as James McCarthur - fresh on the back of signing a new contract - planted a header over the crossbar when picked out by Andros Townsend.
Neither side could take their opportunities when they arrived, which would ultimately prove costly, as Jamaal Lascelles nodded wide from a free header and then failed to make contact at all from the next set-piece delivery.
Milivojevic was failing to make the most of Palace's free kicks, but that almost changed shortly before the interval as he sent a delivery right the way through, with the upright coming to the visitors' rescue.
Having had to ride their luck, Newcastle felt that they should have been awarded a penalty early in the second half when half-time substitute Joselu claimed that he was hauled down inside the box.
Palace were less successful at getting shots in at goal after the restart, while up the other end Joselu's shot hit Tomkins on its way through and was easy for Hennessey to deal with.
The match briefly came back to life with a quarter of an hour to play, as Christian Atsu's shot was kept out by Hennessey and Townsend blasted the ball right at Dubravka from eight yards.
Sakho really should have snatched all three points for his side in the closing stages, only to miss the target from four yards when spotted by Townsend.
There would be one last chance for the Eagles to win it, but McArthur could not quite convert Wilfried Zaha's shot-cross at full stretch, leaving Palace one short of 100 Premier League wins.
CRYSTAL PALACE (4-3-3): Hennessey; Wan-Bissaka, Tomkins, Sakho, Van Aanholt; Milivojevic, Kouyate (Meyer '62), McArthur; Townsend, Zaha, Ayew (Sorloth '70)
NEWCASTLE UNITED (4-4-1-1): Dubravka; Lascelles, Fernandez, Dummett, Yedlin; Diame, Shelvey, Kenedy, Ritchie (Atsu '71); Perez; (Muto '80), Rondon (Joselu '45)