Exeter fought back superbly after conceding two early goals to claim all three points in an entertaining clash.
Chris Todd appeared to put the ball though his own net in bizarre circumstances and Charlie MacDonald made it two from the spot for the visitors midway through the first half.
But a sensational free-kick from man of the match Gareth Sheldon brought Exeter back into the game.
After the break the Grecians dominated and secured victory when Sheldon's corner was helped into his own goal by goalkeeper Phil Smith before substitute Marcus Martin grabbed a winner midway through the half.
Exeter started brightly and striker Jake Edwards headed narrowly over the bar from Barry McConnell's fine cross.
Striker Les Afful looked lively for Exeter early on and had a shot well saved by Smith.
But Crawley stunned Exeter when they took the lead against the run of play in the 11th minute.
A long ball was pumped forward and City were caught napping at the back. Under pressure from Danny Davidson, Todd ended up poking the ball through his own net.
Worse was to come ten minutes later. Adrian Deane was brought down by City defender Dean Moxey inside the box and MacDonald made no mistake from the spot.
Just as all seemed lost Exeter were brought back into the game with Sheldon's spectacular 30-yard free kick.
Sheldon had a header well saved by Smith and then tested the goalkeeper twice more with long range free-kicks.
Exeter were level in the 70th minute. Sheldon's curling corner was met by Smith but, under pressure from Moxey, the goalkeeper appeared to punch the ball into his own net.
Two minutes later they took the lead. Shedon's cross picked out substitute Martin who nodded the ball home from close range.