Walters put the home side ahead before Tuncay, who had missed a host of chances against Tottenham over the weekend, made it 2-0.
Steve Leslie pulled one back for the League Two outfit but Stoke held on to progress to the third round.
Tony Pulis made six changes to the team that was beaten by Spurs at the weekend, with striker Walters keeping his place in the side.
The big talking point, however, was goalkeeper Asmir BegovicÃ¢â‚¬â„¢s decision to refuse to play for Stoke after speculation linking the Bosnian with a move to Chelsea.
Pulis was left disappointed by the decision from a player who only signed last season and the City boss admitted talks over his future would be held tomorrow. Carlo Nash, who signed this summer at the Britannia, started in goal for the home side as his replacement.
Walters, a Ã‚Â£2.75m signing from Ipswich, put Stoke ahead with a header on 26 minutes. Tuncay then doubled their lead 11 minutes later after he headed in from close range.
With just over ten minutes to go, Leslie pulled one back for Shrewsbury to set up a tense finale, but Stoke held on to progress to the next round and record their first competitive win of the season.
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