Reds keeper Phil Smith needlessly brought down substitute Neil Ross to concede a clear-cut spot-kick and hand the Shaymen in a victory they thoroughly deserved.
The visitors took the game to Crawley from the start and went ahead after 15 minutes.
Dean Howell's cross from the left was only half-cleared as far as Martin Foster who lashed home a low drive from the edge of the box.
Foster almost added another just past the half-hour mark with a curling effort which came off the bar before Ryan Sugden had an angled shot well-saved by Smith eight minutes before half time.
Crawley equalised five minutes into the second half thanks to a low drive by John Robinson and could have taken the lead with any one of three chances soon after.
Joe Healy had a point-blank shot saved by Ian Dunbavin, Ian Simpemba miscued a header from two yards out and Allan Tait saw an angled drive blocked by the visiting keeper.
Having ridden the second-half storm, Halifax began to dominate and it was no surprise when they snatched all three points thanks to Smith's moment of madness.