Kewell's Crawley entered the clash rooted to the bottom of the League Two table against a high-flying unbeaten Swindon outfit, but a 36th minute own goal from Oliver Lancashire stunned the home fans at the County Ground.
The visitors then rounded out the win through late goals from Jordan Roberts and captain Jimmy Smith to seal the points for the Socceroos legend's side.
Crawley were under the pump early and goalkeeper Glenn Morris twice denied Luke Norris as the hosts continued to threaten.
But it was the visitors who went ahead when Young's dangerous cross was met by Lancashire, who headed past Lawrence Vigouroux into his own net.
Kewell's men doubled their lead in the 79th minute after Roberts' shot was deflected into the bottom right corner, before Smith converted Dannie Bulman's cross six minutes later.
"I'm proud of them. They (the players) worked hard and worked together," Kewell said after the match.
"We came here with an idea, and they executed it.
"I felt we did enough to control the game and I'm proud of every single one of them. They were outstanding today."
Crawley's win moves them up to 20th on the 24-team League Two ladder, while Swindon slip to ninth.
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