Michael Chopra put the 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 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.
Tuncay had the game's first attempt at goal in the sixth minute, intercepting a stray pass and advancing towards the box before unleashing a shot which David Marshall pushed around the post.
It was a positive start for the hosts, but two minutes later they were behind and only had themselves to blame.
Chopra was given enough space in the box to head a cross against the post and then tuck the rebound past Thomas Sorensen.
The Potters tried to respond quickly and Danny Higginbotham sent a useful delivery into the area which Kevin McNaughton cleared, but they were almost undone by more poor marking soon after as Andy Keogh was allowed a free header which he directed straight at Sorensen.
Jon Walters was next to nod on target at the other end, but he could not beat Marshall.
Tuncay then gave away possession with a misplaced header, Chris Burke picking up the ball only to shoot wide.
Stoke nearly found an equaliser just after the half-hour mark as Wilson's free-kick crashed against the bar, and were back on level terms in the 45th minute as Tuncay collected the ball from Fuller and finished with a low drive from the edge of the area.
Stoke started the second half brightly and the home supporters cried out for a penalty in the opening few seconds when McNaughton appeared to pull back Fuller in the box.
Referee Mark Halsey waved play on but Stoke continued to pressurise, Jermaine Pennant sliding a cross into the danger zone which Fuller let run through to Tuncay.
The Turkey international unleashed a shot that Marshall saved with his legs and Cardiff then survived the succession of corners which followed.
Fuller came close in the 64th minute as he rose athletically to meet Higginbotham's cross, the Jamaican sending his header just the wrong side of the post.
Pulis then introduced a fresh attacking threat by bringing on Etherington for Pennant.
Cardiff managed to get the ball upfield and Sorensen came rushing off his line to halt Chopra's progress before Bluebirds boss Dave Jones threw on Adam Matthews for Burke.
There was another change for Stoke with Jones replacing Walters and Huth got up to meet Etherington's in-swinging corner but could not get much power on it.
Moments later, Tuncay tried to hook the ball to Etherington but Marshall was alert and snatched the ball away.
Glenn Whelan sent an effort over the bar after Etherington's corner had caused panic in the Cardiff box, and Jones then brought a great save out of Marshall after bursting into the area.
Stoke had a flurry of chances to seal victory in the last few minutes but were unable to finish any of them.
Fuller's downward header bounced over Marshall but up onto the bar and Jones sent an effort over from substitute Rory Delap's cross.