The first half was a feisty affair with neither side able to find their rhythm, but Palace should have scored when Jean-Philippe Mateta missed two chances in the opening period.
Leeds survived further late pressure but held on for the draw which takes them to 34 points from 33 matches, five points clear of the relegation zone albeit having played a game more than Everton in 18th. Palace remain 14th on 38 points.
Burnley's recent fine form continued at the weekend as they earned a second successive three points with a 1-0 victory over Wolverhampton Wanderers, giving Premier League stalwarts Everton a lot to worry about as they dropped into the bottom three.
Leeds are not safe yet, but are in a better place to stay up. Their new American coach Jesse Marsch has picked up 11 points in his seven games at the helm, seven more than they managed in previous boss Marcelo Bielsa's last seven in charge.
"It wasn't our best night, however a recipe of not losing and finding a way to pick up points -- we will take the positives," Marsch told the BBC. "We need to find ways of being a bit more dangerous, but a clean sheet and a draw is a big, big positive. The pressure was always going to be big no matter what."
Palace mustered 17 shots in the game, last attempting more without a goal in a Premier League match on Boxing Day 2018 against Cardiff City.
"I am pleased with how we started and finished in the game, but all that was missing was a bit of quality up front," Palace coach Patrick Vieira told Sky Sports. "With a bit of luck we would have scored goals. We created chances, which is important, but we need to finish them off."