Mehdi Kerrouche scored twice as Swindon secured their third consecutive win and sentenced Crawley to their first home defeat of the season.
The visitors took the lead with a bizarre goal when Alan Connell turned home Raffaele De Vita's overhead kick.
Crawley pressed for an equaliser but Kerrouche headed in after Michel Kuipers parried Matt Ritchie's shot.
The Algerian made sure of the win moments later with a shot into the top right corner.
VIEWS FROM THE DRESSING ROOM
Crawley manager Steve Evans told BBC Sussex:
"Its harsh, but that's what happens when you're having the spell we're having. We're decimated with injuries, although we won't use that as an excuse.
"Swindon were efficient, but I think some big decisions haven't gone our way, so we have to grit our teeth, congratulate Paulo, and move onto Saturday.
"I don't think it's a lesson, I just think we're missing huge players. This is our first real test of character in the squad and the management team in 15 months, and we just need to get on the training ground and work through it."
Swindon manager Paolo di Canio told BBC Wiltshire:
"I'm really proud to be a part of this team.
"They [Crawley] are an aggressive team. They have physical presence but they didn't have a clear chance.
"In the second half we built a victory in the perfect way. All the merit and credit goes to the players, they were amazing."