Lewis Chalmers' first goal in six months moved Aldershot 14 points clear at the top of the Blue Square Premier.
Chalmers headed home Ben Harding's inswinging free-kick after six minutes.
Crawley responded well but Jamie Cook's header from Dannie Bulman's free-kick sailed narrowly wide and Kieran Murphy nodded against the post.
A series of good stops by Nikki Bull kept the Shots' in front and they held on despite seeing Scott Davies' penalty saved by Crawley keeper Ashley Bayes.
Aldershot manager Gary Waddock told BBC Southern Counties Radio:
"You don't come to places lie this and walk through the game.
"They threw everything at us, and I just said to the lads to get a clean sheet for a change."
Crawley assistant manger Paul Raynor said:
"There are some bitterly disappointed players in there again.
"We dominated for long periods. We created four or five chances but just couldn't take any.
"But for a little more composure, we could have got a win against the leaders."
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