The 25-year-old winger, who recently spent three months on loan at Luton's Blue Square Bet Premier rivals Grimsby, where he scored twice in 11 appearances, has put pen to paper on a contract at Kenilworth Road until the end of the season.
Crawley manager Ritchie Barker told the club's official website: "Scott has not figured in our plans too much this season and was happy to go out on loan and get some games for Grimsby and did very well there.
"We had a chat last week when his loan spell expired and he's got himself fixed up with Luton Town.
"We thank him for his services to the club and wish him all the best for the future."
Neilson made 69 appearance for Crawley in his two and a half years with the club.