Palace continued to pile the pressure on and almost broke the deadlock through Glenn Murray, but he headed straight at Bogdan from five yards out.
Bogdan kept the scores level again towards the end of the half, using his feet to keep out Zaha's effort after the Palace man had shrugged off a couple of defenders.
Palace were handed the perfect chance to take the lead with ten minutes left when Zaha was upended by Zat Knight in the box. Murray stepped up to take the spot kick and made no mistake to make it 1-0.
The visitors survived a late penalty appeal to claim all three points and move them above today's opponents.