Cresswell, 35, has signed a 12-month deal with the Potters.
"I'm really appreciative of the opportunity to join Stoke City and I'm excited for the season ahead," he told the club website.
"I have been fortunate to work in environments of high standards and professionalism throughout my career. That's something which is ingrained in me and I will deliver every day."
Cresswell made than 350 appearances for West Ham, scoring 11 goals.
"Aaron's career speaks for itself, both in terms of his quality as a player and his character," Stoke sporting director Jonathan Walters said.
"To have played almost 600 games at the level that he has says everything about how well Aaron looks after himself, how reliable he is, and how much trust managers have consistently placed in him throughout his career."