The Premier League club revealed last week that they were monitoring the availability of the 23-year-old and he has now become Stoke's fifth close-season signing, much to the delight of manager Mark Hughes, who spent one season as a player at Camp Nou.
"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," said Hughes.
"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 [on the club's pre-season tour] will help him settle in and will give myself and the coaches the chance to take a good look at him."
Bojan was tipped for big things after making his Barca debut at the age of 16 and scoring 10 goals in 31 La Liga games in 2007-08.
However, he has since failed to reach double figures in a league campaign and had a spell at Roma before re-joining Barca. Loan stints at Milan and Ajax bookended his Barca return.
It was suggested that Bojan would move to the Britannia Stadium on loan, but he has signed permanently.
Deportivo La Coruna and Olympiacos were also linked with Bojan before he opted to test himself in the English top flight.
Stoke chief executive Tony Scholes enthused: "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."