Stoke, who had lost their previous three home games, got off to a confidence-boosting start when Richard Cresswell gave them a fifth minute lead.
The Rangers defence failed to deal with Rory Delap's long throw and Ricardo Fuller pulled the ball back for Cresswell to head in from point-blank range.
Stoke continued to bomb forward with more long throws by Delap and a succession of corners putting the visitors back into their trenches.
John Eustace was inches off target before a clever interchange of passes between Fuller and Mamady Sidibe ended with the former smashing a shot against a post.
A second goal simply had to come and it did in the 20th minute when Liam Lawrence bent in a wonderful 25-yard shot after dangerman Fuller had been tripped.
Rangers almost pulled one back with their first serious attack in the 27th minute when Scott Sinclair's goalbound shot was brilliantly blocked by Cort.
But Stoke's dominance continued and Rangers' night appeared to become tougher when Dexter Blackstock was shown a straight red card for a poor challenge on Ryan Shawcross two minutes after the interval.
The visitors, who felt they were on the wrong end of referee Uriah Rennie's decisions for most of the night, deserved credit because they played much better with ten men than they had with eleven.
And they got their just desserts in the 63rd minute when Rowan Vine seized on a half cleared corner to drive in an unstoppable shot from 18 yards.
Fuller should have settled Stoke's nerves when he fired over an open goal in the 73rd minute, but Cort did just that when he headed in a Lawrence cross four minutes later.