Shea has signed a four-and-a-half year contract at the Britannia Stadium after the club made a successful appeal for a work permit earlier on deadline day.
Stoke boss Tony Pulis told the club's official website: "It's fantastic news that we have been able to push through a deal.
"Brek is a very, very exciting player, and one who we firmly believe could go on to become an outstanding player for this football club.
"He has terrific pace, a very good left foot and is capable of scoring goals. He has all the tools needed to be a success here."
Shea is the Potters' first signing of the January transfer window. The club also confirmed that Rory Delap has agreed a loan move to Barnsley until the end of the season.