The 19-year-old Kingston-upon-Thames-born Republic of Ireland Under-18 international has signed for an undisclosed fee, understood to be in the region of £250,000.
Having passed his medical, Sweeney has signed a three-year deal.
Sweeney made his Dons debut nine days before his 18th birthday in April 2015.
Since then he has taken his number of first-team appearances to 13 for his home-town club - Wimbledon play their home games in Kingston.
Sweeney signed an undisclosed-length new contract with Wimbledon in March after first being linked with Premier League clubs Tottenham and West Ham.
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Stoke City have completed their third summer transfer deal by signing teenage defender Ryan Sweeney from promoted League One side AFC Wimbledon.