Scott Arfield's missed penalty denied Burnley their first Premier League win of the season. Eagles goalkeeper Julian Speroni palmed Arfield's late spot-kick to safety to deny Sean Dyche's men their first victory of the campaign.
Toothless finishing from the Eagles left Neil Warnock's side still seeking their first league win of the new term too, with Scott Dann heading against the bar and Dwight Gayle fluffing two fine chances.
Burnley captain Jason Shackell was forced to clear off the line after almost putting through his own net, and Jonathan Williams and Jason Puncheon both failed to convert promising opportunities for the hosts.
The Clarets will sweat on an injury to striker Danny Ings, who was forced out of action midway through the first half.
Palace should have put this clash to bed by half-time at Selhurst Park, but wound up wiping collective brows at the stalemate after Arfield's penalty miss.
While the visitors were for the most part devoid of ideas in the final third, Palace wasted a host of chances in either half.
New boss Warnock can take credit for restoring a modicum of the stability that was such a feature under Tony Pulis, but Burnley will need far more craft for Premier League survival.
Neither side could have whinged had Palace taken a 3-0 lead into the break.
Instead the game remained deadlocked at the turnaround, with the first half petering out despite a lively start. Ings' penalty claims ended in a yellow card for the Burnley striker.
Sunday Indo Sport