The contest took place just hours after Palace confirmed that Roy Hodgson had stepped down from his post as manager, with Austrian boss Glasner appointed as his successor on two-and-a-half year deal.
Glasner was in the stands as Paddy McCarthy and Ray Lewington led the Eagles, who broke the deadlock midway through the second half courtesy of a superb long-range strike from Jordan Ayew.
However, Palace were unable to hold on for three precious points as Onana headed home Dwight McNeil's corner kick five minutes from time to ensure the spoils were shared on Merseyside.
A draw for Palace extends their winless league run to three matches and sees them remain in 16th place in the Premier League table, five points ahead of Sean Dyche's Everton, who have climbed out of the relegation zone and up to 17th spot above Luton Town, who have dropped into the bottom three only by goal difference.
The headline team news from Everton's perspective prior to kickoff was the return of Abdoulaye Doucoure to the starting lineup following a spell on the sidelines through injury, with Jack Harrison dropping to the substitutes' bench as a result.
As for Palace, Sam Johnstone replaced Dean Henderson between the sticks after eight games on the bench, while forward Odsonne Edouard - starting his first league game in two months - and defender Joel Ward returned at the expense of Matheus Franca and the injured Marc Guehi.
Glasner was not involved on the touchline or in the dressing room for Palace, but it quickly became clear that the Eagles set up with a 3-4-3 formation, one used regularly during the Austrian's time in charge of former club Eintracht Frankfurt.
A cagey and at times scrappy start was made by both teams in the opening exchanges, but it was Palace who looked the most assured in possession and tested the gloves of Jordan Pickford for the first time in the 12th minute when Edouard fizzed a low-driven shot from a tight angle.
Everton responded well as Doucoure fired a strike from the edge of the penalty area a few yards wide, before Dominic Calvert-Lewin thumped a towering header off target after meeting an inviting cross from McNeil on the left flank.
Just after the half-hour mark, a well-worked free-flowing move from Palace resulted with Daniel Munoz teeing up Jefferson Lerma for a first-time shot from just inside the box, but he was off balance and leant back slightly as he placed his effort high over the crossbar.
Everton's Ashley Young was then required to be switched on at the back post as he cleared Jean-Philippe Mateta's downward header off the line from a Palace corner.
The contest came to life in the final five minutes, but there was a lack of quality on show as both sides took it in turns to break away on the counter-attack and scrambled the ball to safety, much to the frustration of the Goodison Park faithful.
Clear-cut chances were at a premium at the start of the second half, but Palace were presented with a decent opportunity to break the deadlock on the 56-minute mark when Mateta slipped the ball through for Tyrick Mitchell on the left side of the penalty area, but the left-back was closed down well by Pickford, who managed to beat away the attempted chip.
Everton then came close at the other end just after the hour mark when Johnstone palmed away James Tarkowski's powerful header from a corner before Doucoure tapped an effort towards goal at the back post which was swatted away by Johnstone.
Dyche turned to Onana and Harrison to provide a spark off the substitutes' bench, but just moments after their introduction, Palace opened the scoring out of nowhere with a stunning long-range strike from Ayew in the 66th minute.
A long ball forward was held up and laid off by Mateta to Ayew, who took one touch before arrowing a powerful 20-yard strike beyond a helpless Pickford into the far corner to silence the home supporters.
That goal sparked Everton into life heading into the final 15 minutes, a period in which only basement club Sheffield United (19) have conceded more goals than Crystal Palace (16) in the Premier League this season.
Calvert-Lewin headed narrowly wide before Johnstone dived to his left to palm away a half-hearted shot from Garner, but the Toffees managed to restore parity in the 85th minute when Onana towered above the Eagles backline to nod home McNeil's corner from just a few yards out.
Everton were unable to find a winner in stoppage time, and while they can take positives from claiming a point against Palace, Dyche's men have now gone eight league games without a victory.
The Toffees will continue to search for their first win of 2024 when they travel to Brighton & Hove Albion on Saturday, while Glasner will be in the dugout for his first Crystal Palace match at home against relegation-threatened Burnley.