The 27-year-old Scottish international has put pen to paper on a two-year deal with Blues.
Glasgow-born Burke has previously played for manager Alex McLeish during his spell with Rangers.
He joined Cardiff midway through the 2008-09 campaign but has now left the club after failing to agree a new contract.
Burke played in the Champions League during his spell at Ibrox and tasted Scottish Premier League and League Cup success during a nine-year stay at his boyhood club.
He was an integral part of a Cardiff side that came close to achieving promotion to the top flight on several occasions.
His two Scotland caps and goals both came in the Kirin Cup in Japan in 2006.
Source: PA