Preston soaked up early pressure and began to gain confidence before Stoke bounced back to take a 16th minute lead.
A trademark long throw by Rory Delap caused the damage with Cort rising to glance a header which flicked the far post on its way in.
Danny Pugh then wasted a chance to extend the lead before a dreadful mistake by Preston keeper Andy Lonergan handed City their second goal in the 27th minute.
Lonergan inexplicably spilled a tame cross from Liam Lawrence and Richard Cresswell pounced to score against his eighth goal of the season against his former club.
The howler flattened Preston for a spell and it could have been game over before half-time as Stoke continued to pile forward against the visitors' uneasy defence.
A similar pattern followed at the start of the second period and Lonergan partly atoned for his error when he made a terrific save, after another Delap throw was headed towards his own goal by Youl Mawene.
But Preston began a spirited fightback and Chris Brown went close and then deservedly reduced the arrears in the 68th minute.
Former Stoke winger Lewis Neal headed Simon Whaley's cross back across goal and Brown prodded the ball home from point-blank range.
It looked game on at that point, but it took Stoke just four minutes to restore their two-goal cushion as Cort headed his second of the match and sixth of the season.
Lawrence swung over a right-wing corner and Cort rose majestically to power in his attempt which struck the underside of the bar before crossing the line.