Mile Jedinak had an early chance for the Premier League outfit, but the midfielder could only fire his effort high and wide of the crossbar.
It was Crawley who took the lead just before the half-hour mark, however, when Clarke pounced on a loose ball inside the box before tucking it past the onrushing Julian Speroni.
Jedinak and Wilbraham both had half-chances for Palace in the latter stages of the half, but neither could find the back of the net from close range.
Palace continued to threaten in the early stages of the second period with Jedinak and Quincy Owusu-Abeyie both seeing efforts go wide of the post.
Yannick Bolasie was next to come close for the newly-promoted Premier League outfit just before the hour mark, but his effort was wide of the target.
Palace continued to dominate as the match reached its latter stages, with Peter Ramage heading over from close range.
However, it was Crawley who wrapped up the win in the latter stages when Sergio Torres converted from close range after James Hurst had been denied inside the box. Mat Sadler added their third in the final minute.