Brendan Rodgers' side will remain second overnight with Everton, level on points with them, not playing later on Monday after their home match against Manchester City was postponed following more positive COVID-19 cases at City.
Leicester dominated the bulk of the game but failed to take their chances and missed a first half penalty while striker Jamie Vardy only came on as a second half substitute.
"We're disappointed not to get three points and we weren't as clinical as we have been, but it's onto the next game," Barnes told Amazon Prime.
"It has been a very busy schedule with some great results so we're happy with where we are in the table."
Leicester could have taken an early lead after Luke Thomas was fouled in the box by James Tomkins, but Kelechi Iheanacho's weak effort from the spot after a stuttering run-up was swatted away by goalkeeper Vicente Guaita.
"Guaita is a good keeper, he's been brilliant ever since he came here and he saved me today. Hopefully I can repay him one day," Tomkins said.
Palace made it 1-0 just before the hour mark when Andros Townsend floated a cross to the far post for Zaha to volley home from an acute angle.
However, Barnes equalised when he scored his second goal in as many games, driving a low effort from the edge of the box past the outstretched arm of Guaita in the 83rd minute.
Palace's winless run in the league has now extended to five games but Zaha was glad to take a point.
"We'll take a point today, the last few games have been hard on us," he said after scoring his eighth league goal this season.
"The way we worked today I'm definitely happy and we'll take that going into the new year."