Torquay leapfrogged Crawley in the League Two play-off places as Eunan O'Kane grabbed the only goal of the game at the Broadfield Stadium.
The Irish midfielder secured an eighth away win for the Gulls from close range following a cross from Rene Howe.
Torquay, who had Danny Stevens carried off, could have gone behind just 10 seconds into the game but Robert Olejnik blocked Billy Clarke's shot.
And the keeper denied Clarke again by diving to save another goalbound drive.
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Crawley assistant boss Paul Raynor told BBC Sussex:
"It was not a typical Crawley performance. We lacked the normal tempo and desire, and played into Torquay's hands.
"Sometimes you've got to take it on the chin and be honest. We lacked cohesion, we looked disjointed, and we need to look long and hard at it.
"That certainly was unacceptable for a Crawley Town that we send out, that plays with purpose and passion and needs options on the ball. To say we're disappointed is an understatement. We just weren't good enough."
Torquay United manager Martin Ling told BBC Radio Devon:
"I thought Rene Howe was outstanding today. He's a battering ram that's got skill as well.
"Olejnik is the best goalkeeper in the bottom two leagues, in my opinion.
"All 12 players we used today were immense."