Sean Dyche's side were in danger of a damaging defeat in the fight for Premier League survival after Jordan Ayew blasted Palace ahead with a fine finish in the second half at Goodison Park.
But Onana levelled with six minutes left to lift Everton out of the relegation zone above Luton on goal difference. If Palace had held on, they would have moved eight points clear of the bottom three, but instead the gap to Luton is five points.
It was a blow to former Eintracht Frankfurt boss Glasner, who was appointed by Palace after Roy Hodgson stepped down earlier on a dramatic day for the south London club.
Hodgson's assistants Paddy McCarthy and Ray Lewington were in charge against Everton, but Glasner was seated alongside Palace chairman Steve Parish and sporting director Dougie Freedman in the Goodison stands.
"It was a wonderful goal by Jordan Ayew, but we couldn't hold on. There's lots of quality and Glasner is coming into a squad with plenty of character," McCarthy said. "I'm sure the players were aware he was watching. There are lots of positives he will have seen and I'm sure he cannot wait to get to work."
Everton boss Sean Dyche added: "We let in a bad goal from our point of view, but there was a lot of good in that second half. Our mentality has been key the last few weeks and will be going forward."
Glasner has agreed a contract until June 2026 and will be in charge for the first time when Palace host second bottom Burnley on Saturday in another crucial clash. He faces a tough task to revive a team with only three wins in their last 20 games in all competitions during an alarming slide towards the relegation zone.
Former England and Liverpool manager Hodgson's departure was suitably respectful for a boyhood Palace fan who
Read on besoccer.com