Although Begovic is now a Bosnia-Herzegovina international, he played for Canada at the FIFA under-20 World Cup in 2007. He was co-captain with Will Johnson, now captain of Major League Soccer's Portland Timbers.
Chelsea says Begovic has signed a four-year contract. The transfer fee wasn't disclosed.
Petr Cech joined Arsenal last month after becoming disillusioned at being the second choice to Courtois last season.
Begovic is Chelsea's second signing of the off-season, following the arrival of striker Radamel Falcao on a season-long loan.
Born in Yugoslavia in 1987, Begovic was four when his family fled war-torn Bosnia. After seven years in Germany, they moved to Edmonton.
He attracted attention when he was 15, making the provincial and national team. An agent helped set up trials in England with Portsmouth and Tottenham. Begovic liked Portsmouth so much that he signed before trying out with Tottenham.
He was loaned out to Belgium's La Louviere and England's Macclesfield Town, Bournemouth, Yeovil Town and Ipswich Town before signing with Stoke City in early 2010.
He made 172 league and cup appearances for the Potters after joining the club from Portsmouth in a 3.25-million-pound transfer.