Palace were determined to stop any late drama from going against them, and carved out bright chances when Olise tried to feed Jean-Philippe Mateta and Clyne sought to square to Mateta, Zaha and Schlupp yards from goal.
Burnley handled both chances to keep the game level - and the clock ticked on further.
The south Londoners looked bright throughout five minutes of added time but couldn't repeat their ninth-minute efforts, and ultimately had to contend with a point from a fairly level clash with in-form guests.
Palace: Butland, Mitchell, Guéhi, Andersen, Clyne, McArthur, Milivojević (Mateta 60), Gallagher, Schlupp (Kouyaté 85), Zaha, Olise.
Subs not used: Guaita, Kelly, Hughes, Eze, Ayew, Edouard, Benteke.
Burnley: Pope, Roberts, Tarkowski, Mee, Pieters (Taylor 60), Lennon, Cork (Westwood 60), Brownhill, McNeil, Weghorst, Rodriguez (Barnes 88).
Subs not used: Hennessey, Lowton, Collins, Long, Bardsley, Cornet.