Michael Chopra put the npower Championship side in front in the eighth minute, the Potters defence allowing him time to head against the post and then slot home the loose ball.
Tuncay equalised on 45 minutes after latching onto Ricardo Fuller's header, but although the hosts dominated the second period, the Bluebirds managed to keep their Barclays Premier League opponents at bay.
Stoke boss Tony Pulis rang the changes, with Tuncay, Marc Wilson, Robert Huth and captain Ryan Shawcross the only players retained from the XI which started Tuesday's 2-1 defeat at Manchester United, and the likes of Kenwyne Jones and Matthew Etherington dropping to the bench.
Cardiff were missing some big names too, though, most notably top scorer Jay Bothroyd and on-loan Manchester City forward Craig Bellamy, neither of whom featured in the squad.
Stoke had a flurry of chances to seal victory in the last few minutes but were unable to finish any of them.