However, after numerous missed opportunities, Frank de Boer could now reportedly be removed from his position after Palace became the first top-flight team to lose each of their opening four matches without scoring a goal since Preston North End in 1924.
It took Burnley less than three minutes to take the lead in the North-West but it was gifted to them by Lee Chung-yong, whose misplaced back-pass set up Wood to curl the ball past a stranded Wayne Hennessey from 30 yards out.
However, to their credit, Palace could - and probably should - have netted an immediate equaliser, with a Scott Dann half-volley being cleared off the line by Stephen Ward before Christian Benteke directed a header marginally wide from 12 yards out.
Back up the other end, Sam Vokes glanced a header wide of the far post in what had quickly developed into a frantic contest, but when the action began to calm down it was Palace who gained control.
James McArthur should have done better with a free header after meeting a corner while Yohan Cabaye wasted a chance from a free kick, but the Eagles' rhythm was disrupted when Burnley goalkeeper Tom Heaton was forced off with what appeared to be a dislocated shoulder.
Palace continued to have the better of the play after the break, but neither side were able to create anything of note in the third quarter of the game.
It was proving to be a frustrating period for De Boer and the Dutchman soon decided to introduce Levi Lumeka for his debut, with the youngster taking just a few minutes to go close to the perfect introduction.
Townsend's cross towards the back post found the newcomer, but Lumeka's header was deflected over, with Jeffrey Schlupp drilling over from 15 yards from the resulting corner.
Burnley had offered very little going forward but Ashley Barnes almost doubled the home side's lead when he directed a curling effort towards the far corner, only for Hennessey to dive to his right to palm the ball away.
The closing stages belonged to Palace - who needed to throw caution to the wind - and they could count themselves unfortunate to not find the back of the net with at least three openings.
Firstly, Benteke was denied by a brilliant save from close range from Pope, before Dann saw another half-volley excellently cleared off the line after making a clean connection from 12 yards out.
Townsend and Cabaye both missed out with long-range efforts, but the chance of the game went to Dann, who headed wide from three yards out after being picked out by Cabaye.