Neither side broke the deadlock in the first half, although Jonathan Forte came close on two occasions, forcing a save from Glenn Morris before nodding just over the bar.
The visitors were forced to hang tight in the second half, although there was hope for them when Curtis Thompson was sent off in the 72nd minute.
Nonetheless, Kevin Nolan's charges took the lead not long after when Thierry Audel forced the ball over the line when Rob Milsom's corner had caused pandemonium inside the box.
It was no more than Notts deserved after creating numerous chances, with Forte spurning two in the first half.
Crawley, however, hit back in the final 10 minutes when James Collins had space outside the box and cannoned in a terrific effort off the underside of the crossbar from 25 yards.
That had seemingly curtailed the Magpies' hopes of victory, but right at the death, Forte scored on the rebound after Haydn Hollis's original shot had been saved.
Share your thoughts about the match on Pride of Nottingham by signing up to the website, visiting the forum and joining the conversation.