The sides couldn't be separated after a low-key 90 minutes at St. James' Park, with neither side able to make a breakthrough.
The tie went straight to penalties, with goalkeeper Pope taking centre stage with a hat-trick of saves from the spot to deny Luka Milivojevic, Jean-Philippe Mateta and Malcolm Ebiowei and hand Eddie Howe's side what was Newcastle's first-ever penalty shoot-out victory on their own patch.
The visitors made the brighter start on Tyneside, with plenty of early running failing to bring any real, clear-cut scoring opportunities. They frustrated United well, though, with Pope forced into the first save of the night to deny Mateta. Will Hughes' weaving run and smart through ball created the opening and the Eagles' frontman struck it well, but Pope did well to repel it.
The United stopper will hope to be part of England's World Cup squad when it is announced on Thursday, but he didn't have too many other chances to impress in the first half as Palace struggled to muster another effort on target. James Tomkins headed a corner over but the other noteworthy efforts came from Howe's men - Elliot Anderson with a smart turn and shot that was easily held by Sam Johnstone, and a speculative Jonjo Shelvey free kick from close to the halfway line which floated just wide.
Within seconds of the restart, Tomkins' weak backpass almost gifted United an opener. Chris Wood, in for Callum Wilson up front and a scorer in Sunday's 4-1 win at Southampton, pounced on the loose ball, but Johnstone stood up to deny the former Burnley forward from a tight angle.
Jordan Ayew curled a shot well wide soon after while at the other end, Anderson saw a header deflected past the upright and Dan Burn - one of only three starters who lined up for that weekend win at St. Mary's - planted a header straight at Johnstone after an excellent clipped cross from Allan Saint-Maximin picked him out in the box.
Howe sent on Bruno Guimarães, Kieran Trippier and Sven Botman with just under 25 minutes remaining and Miguel Almirón for the final 15, and the changes gave the hosts an injection of purpose. Almirón saw a low effort parried by Johnstone just after his introduction, Joelinton whipped one off target from distance and Matt Targett thumped over as Newcastle finished the contest firmly on top.
But with a goal not forthcoming and the two sides inseperable, penalties was required to decide the outcome. Wood, Trippier and Joelinton all converted with ease for United, with Hughes and Ward converting for Patrick Vieira's men.
But despite Botman's miss and Guimarães blazing his effort over the bar, another conversion from the hosts wasn't required as Pope - who had earlier produced terrific stops from Milivojevic and Mateta - kept out substitute Ebiowei's weak attempt to create Magpies history and send United through to the fourth round.
