The 31-year-old central defender has agreed terms on a two-year contract and will join Scunthorpe once his current deal with Oxford expires.
A robust and capable defender, Crosby has made over 450 appearances in his career and was recently named in the PFA Third Division team of the year.
Laws is clearly delighted with his latest acquisition and has predicted more faces to follow Crosby to Glanford Park.
''We are obviously delighted to have the first part of the jigsaw,'' Laws told the club's official website.
''Andy Crosby is a very experienced centre half, a very good leader and is a big, strong boy.''
