The two teams could not be separated during the 90 minutes, but penalties from Chris Wood, Kieran Trippier and Joelinton proved enough for the Magpies to win 3-2 in the shootout, with Luka Milivojevic, Jean-Philippe Mateta and Malcolm Ebiowei missing for the Eagles.
James Tomkins had the first half-chance of the match in the 11th minute, but the centre-back's header was always rising over the Newcastle crossbar.
Palace had another opportunity in the 22nd minute when Mateta found space in a dangerous position before striking towards goal, but Magpies goalkeeper Nick Pope was on hand to make a smart save.
Sam Johnstone was then called into action down the other end to keep out a long-range effort from Elliot Anderson, with the two sides still level approaching the end of the first period.
Quick thinking from Jonjo Shelvey almost caught Johnstone out in the 33rd minute, with the midfielder attempting to score a free kick from distance, but his effort dropped wide, and a relatively uneventful first period ended with the two sides level at 0-0.
Wood had an opportunity to send Newcastle ahead early in the second period, but the striker could not beat Johnstone from a tight angle, before Dan Burn headed wide from close range in the 55th minute.
Pope kept out an effort from Will Hughes in the 59th minute, but Newcastle were the better side at the start of the second half, with Eddie Howe's side putting pressure on the Palace backline.
There was to be no goal inside 90 minutes, though, with the match heading for a penalty shootout.
Sven Botman and Bruno Guimaraes missed for the Magpies after Wood, Trippier and Joelinton had converted, but Palace were only successful from two of their five attempts, and Ebiowei's miss proved to be crucial, as the home side progressed to the fourth round.