The 22-year old winger completed his £2.5million move from FC Dallas on transfer deadline day and is looking forward to beginning his four-and-a-half-year stint at the Britannia Stadium.
"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 Potters' official website.
"It's one of the best leagues in the world, and I'm excited to be here and hopefully I can contribute."
Dallas head coach Schellas Hyndman said it was unfortunate that Shea had departed Frisco but wished the USA international all the best.
"This is one of those bittersweet situations," Hyndman said in a statement. "We are very excited for Brek to take this important step in his career, but we're also going to miss him here.
"This will be a good, fresh challenge for Brek, and we believe he is well-prepared and ready to make an impact for Stoke."