The 23-year-old, who broke into the Barca first team in 2007 but struggled to make an impression after re-signing following two years at Roma, moves to the Britannia Stadium for an undisclosed fee.
A statement issued through Stoke's official Twitter account read: "£HolaBojan Stoke City are delighted to confirm the signing of @BoKrkic from @FCBarcelona on a 4-year deal".
Bojan, who becomes Stoke's fifth signing of the summer in following Mame Biram Diouf, Steve Sidwell, Phil Bardsley and Dionatan Teixeira to Staffordshire, will fly out to join his new team-mates at their pre-season training camp in Reit im Winkl, Germany.
Potters manager Mark Hughes told stokecityfc.com: "Anyone who knows European football will be aware of him as a player and the fact that he sees his future at Stoke City is really exciting and an endorsement of the club itself.
"He's desperate to prove himself in the Premier League and we're pleased to be able to offer him that platform.
"The time he spends with the squad in Germany will help him settle in and will give myself and the coaches the chance to take a good look at him."
Bojan is the son of former Red Star Belgrade midfielder Bojan Krkic Sr, the Serbian who became Barcelona's head scout in 2007 and held the position for four years.
He promoted his son's graduation from the famed Barca academy to the first team in September 2007 and Bojan went on to score 27 league goals in over 100 appearances for the Catalan club before joining Roma in 2011.
After failing to make a significant impression in the Italian capital he spent a season on loan with AC Milan before returning to the Nou Camp in the summer of 2013.
He was immediately loaned out to Dutch club Ajax and helped the Amsterdam outfit secure the 2014 Eredivisie title by scoring five goals in 32 games.
Stoke chief executive Tony Scholes said: "We are delighted to have signed such a richly talented and exciting player, who is still only 23.
"A number of other clubs had expressed an interest in him but Bojan believes joining Stoke City will benefit his career and we have no doubts he will prove to be a wonderful addition to our squad.
"The signing of Bojan, along with the other players we have recruited ahead of the new season, underlines the progress we are making as a club."