Stoke should have taken the lead earlier when Allen headed an easy chance wide but had their patience and persistence rewarded when Ramadan Sobhi laid the ball off to the Welshman, who steered in from 10 metres on 67 minutes.
Palace sent on Jeffrey Schlupp to join other new signing Luka Milivojevic but only looked truly dangerous from set-pieces. They remain 19th, above bottom-placed Sunderland on goal difference.
(Reporting by Neil Robinson; Editingt by Toby Davis)
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