Eagles winger Brennan Johnson had a chance to boost the survival hopes of his former side Tottenham but missed a free header in the first half on Monday night (Tuesday AEST).
The visitors were not without chances to grab all three points, and it took a big Dean Henderson save to deny Greek centre back Konstantinos Mavropanos before the break.
The result lifts the Hammers two points clear of 18th-placed Spurs, while Palace – who have a game in hand – move within five points of Brentford, who currently occupy seventh and a Conference League play-off round position.
Nuno Espirito Santo named an unchanged West Ham side from Friday's victory over Wolves.
Johnson had a big chance to open the scoring from an excellent Jefferson Lerma cross in the 20th minute but failed to angle his header on target, then Henderson kept out a Crysencio Summerville free kick.
Johnson, teed up by Yeremy Pino, fired his next attempt wide, and Palace breathed a sigh of relief after they failed to clear a corner, allowing Axel Disasi a shot from point-blank range, though the offside flag would have been raised.
There were just over six minutes remaining in the half when Maxence Lacroix came in with a big block to deny Taty Castellanos.
Henderson had just punched away Jarrod Bowen's header and the ball made its way to the Argentina international, who attempted a bicycle kick, but Lacroix got his body in the way just in time.
It remained goalless at the break after Henderson reacted quickly to palm away Mavropanos' sharp header and Glasner made a triple substitution just before the hour.
Lerma drew Hermansen into a straightforward save with a header and the stalemate remained intact after a slapstick series of slip-ups by both sides at the other end of the pitch.
The visitors won an added-time corner but it was eventually cleared by Palace to ensure the points were shared.
