Miles Addison marked his first start after 12 months of injury hell by cancelling out Craig McAllister's first-half opener with a 62nd-minute header at the Broadfield Stadium.
But Derby, who missed a penalty through Kris Commons, saw their hopes of a replay snuffed out in stoppage time when Torres converted Dean Howell's corner to put the Blue Square Premier side through to a fourth-round tie at Torquay.
Commons spurned the perfect chance to fire Derby ahead in the ninth minute when his tame penalty was saved by Michel Kuipers after the Dutch goalkeeper was adjudged to have brought down Chris Porter.
The Rams were punished 20 minutes later when, with a strong wind at his back, Kuipers sent down a huge clearance that caused chaos in the Derby box and the ball fell to McAllister who fired the Red Devils ahead.
Crawley were lucky to reach the break with 11 men on the field when sought-after striker Matt Tubbs escaped with a booking for a two-footed lunge on Robbie Savage.
Derby were back level after 62 minutes when Commons went some way to atoning for his earlier miss by curling in a wicked free-kick for Addison to head home only for Torres to strike at the death.