The 22-year-old was granted a work permit after a successful appeal at a Wembley hearing, which sealed his move from FC Dallas.
"It's fantastic news that we have been able to push through the deal," said Potters boss Tony Pulis.
"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 had already passed a medical with Stoke but the deal was held up as he does qualify for a work permit under current regulations.
Shea was required to have played in 75 per cent of the USA's international games in order to be awarded a permit, but he has only featured in 14 games over the past two seasons.
However, that was largely due to Shea undergoing an operation last August to remove a bone in his right foot, forcing him to miss several internationals, and Stoke successfully argued mitigating circumstances.
"It's been up and down, trying to figure out what's going on, but I'm here now and ready to go," said Shea.
"It's one of the best Leagues in the world, and I'm excited to be here and hopefully I can contribute."