The former FC Dallas man had seen his move held up by his failure to secure a work permit, although he was granted one by an appeal panel.
Tony Pulis travelled to Wembley on deadline day in order to convince the panel of Shea's abilities, and the club has now confirmed the £2.5 million capture of the 22-year-old.
"It's been up and down, trying to figure out what's going on, but I'm here now and ready to go," Shea told the club's official website.
"It's one of the best Leagues in the world, and I'm excited to be here and hopefully I can contribute."
Shea, who has represented United States 15 times in his career, has been tracked by the Potters for the majority of the window, with the winger initially joining on loan at the start of January, and Pulis is delighted at finally cementing the move.
"It's fantastic news that we have been able to push through the deal," Pulis added.
"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."