Leading through an Alexis Sanchez header, the Gunners would have regained third spot in the Premier League had forward Yannick Bolasie not conjured a late leveller for Palace.
But a second successive draw left Arsenal fourth, behind third-placed Manchester City on goal difference, and just four points ahead of Manchester United in fifth.
“Honestly, after the game, I am not in that kind of mood to dream about the [Premier League] championship, I am more in the kind of mood to repair mental damage and prepare for the next game,” said Wenger.
“It’s much more about that and to look behind us because everybody is playing well and winning games. For us, it will be a fight until the end to get a place in the top four. If we can do more we will do more, but let’s not dream, let’s focus on the next game.”
In an opening period characterised by Arsenal possession without penetration, the shot-shy Gunners failed to create a clear-cut chance until the 43rd minute when Mesut Ozil was denied by goalkeeper Wayne Hennessey.
Eagles striker Connor Wickham cleared the crossbar with a free header moments before Sanchez darted onto Danny Welbeck’s dinked pass to nod over the advancing Hennessey in first-half stoppage time.
Palace equalised inside the final 10 minutes when Bolasie’s effort from the edge of the penalty area, following a lung-busting run from ex-Gunner Emmanuel Adebayor, scythed through the clutches of Cech.
The leveller leaves Arsenal on 499 Premier League wins and just four points clear of the chasing pack as their hopes of a 20th consecutive top-four finish remains in the balance.