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.
