Crawley let slip a promising three-try lead before complacency set in allowing Brighton to recover and win.
Confident Crawley produced three tries in the opening 23 minutes scored by Richard Hewitt, Andy Evans and Rob Gilliat.
They could have added two more easy tries in the first half but the execution was overcomplicated and wasted.
Brighton rallied and attacked the Crawley line with vigour.
Sustained pressure put Brighton over the line between the posts for an easily converted try just before half-time.
With the second half starting 7-15 there was still all to play for.
An early penalty for Brighton brought them to within five points of the leaders and Crawley now started to fall foul of the whistle on too many occasions.
In the second half Brighton crossed the line to tie the game and the conversion took them into the lead.
Disallowed tries for Dan Keegan and Ross Keeley would have put Crawley back on track but as the game drew close to the end it was Brighton who looked more in control and with just minutes on the clock scored another try.
Crawley missed 12 points from the kicking tee showing that opportunities must be taken when playing leading rivals.
This Saturday Crawley are at home to Seaford.
Crawley: Theobald, Mansfield, Wrigley, Gosling, Gilliat, Dean, Morgan-Smith, Thompson, Searle, Spurgeon, Hewitt, Bowen, Evans, Keeley, Cockram, Ferrier, Plunkett, Keegan