Tony Pulis' men missed the chance to open a three-point gap at the top, but not even he could argue his misfiring side deserved to do so.
Blackpool's late arrival after a serious accident on the M6 forced a 30-minute delay to the kick-off.
And Stoke set out to take advantage with Salif Diao firing just over in the first minute and Mamady Sidibe heading Paul Gallagher's fifth minute free-kick against the outside of a post.
But the visitors soon settled to create some problems of their own as Carlo Nash saved from Gary Taylor-Fletcher and Claus Jorgensen in quick succession.
Stoke were lacking a sting in the tail without the suspended Ricardo Fuller, as his deputy Richard Cresswell also struggled to retain possession.
Blackpool survived a 36th minute scare when Paul Rachubka saved well from Gallagher, but the Seasiders stunned the home fans by taking the lead just a minute later.
Ben Burgess was allowed too much space to tee off a 16-yard shot and the ball took a vital deflection off home defender Cort to leave Nash helpless.
The visitors almost extended their advantage in first-half stoppage time when Ian Evatt steered his shot wide from close range, after Stoke failed to clear a corner.
But Stoke were back on terms less than two minutes into the second half, getting a helping hand from Blackpool scorer Burgess.
The big striker headed Gallagher's corner straight to Cort who buried his eighth goal of the season from six yards, his shot taking a deflection on its way in.
Stoke's expected stampede never materialized after that, however, with Blackpool looking every bit as likely to take the three points.
And the visitors deserved to survive an injury-time scare when Jorgensen cleared Gabriel Zakuani's header off the line.