The 22-year-old Scotland international defender has moved to St. Mary's for a fee estimated at £500,000 and has signed a three-and-a-half year contract.
Crainey has been unable to command a regular place at Parkhead and has made only 6 appearances for Martin O'Neills side this season.
"We are a little short of left-backs and that is a difficult position to fill. He is the right age and will be a good one for the future," chairman Rupert Lowe told the clubs official website.