The full-back headed Matthew Etherington's corner into his own net under pressure from Robert Huth to complete a concerted fightback by the Potters.
Kevin Nolan's 43rd-minute penalty had given the Magpies a deserved half-time lead, but Stoke responded after the break and were back on terms with 67 minutes gone when Sunderland old boy Kenwyne Jones, who had earlier hit both the post and the bar, headed home from close range.
But there was still a sickening twist to come five minutes from time when Perch's unwitting intervention handed the visitors victory in front of a crowd of 41,915.
Last Saturday's 1-0 Premier League win at Everton and their midweek Carling Cup heroics at Chelsea had made for a positive week for the Magpies, who returned to home turf determined to erase the memory of their last outing on Tyneside.
A fortnight ago, they turned up to face Blackpool with nothing on their minds but taking three precious points, but ended up empty-handed after Ian Holloway's men left with a 2-0 win under their belts.
Despite their last two results, both achieved by impressive performances, there was no room for complacency against a Stoke side whose prowess from set-pieces - and in particular Rory Delap's long throw - is well-documented.
Newcastle will not have been at all disappointed to see Delap's name among the substitutes and without him, to say the Potters were one-dimensional before the break would have been something of an understatement.
Plan A appeared to be to get the ball up to Jones as quickly as possible, but with the former Sunderland frontman having the kind of afternoon which persuaded Black Cats boss Steve Bruce to cash in on him this summer, the threat was minimal.
By contrast, the Magpies were enterprising going forward with the insurance provided by defensive midfielder Cheik Tiote allowing Joey Barton to run the midfield, while winger Hatem Ben Arfa proved a constant source of menace.
For all that, Chris Hughton's men were able to create few clear-cut chances with Ben Arfa's 19th-minute free-kick, which curled just wide, as close as they came to opening the scoring in the early stages.
The Frenchman's combination with full-back Jose Enrique down the left looked the most likely avenue for the home side and the Spaniard drilled a low cross across the face of goal seconds later.
But it was from that flank that the opening goal did arrive, albeit indirectly as Stoke's stubborn resistance was finally broken.
Enrique's towering cross attracted Andy Carroll's attention, although it seemed destined for the arms of keeper Thomas Sorensen.
However, before the Dane could get there, Huth barged the striker to the ground and, to his horror, looked round to see referee Mike Jones pointing to the spot.
His prolonged protests went unanswered and Nolan stepped up to face Sorensen, who once saved an Alan Shearer penalty at St James' Park during his Sunderland days, from 12 yards.
But there was to be no repeat as the Magpies' skipper sent him the wrong way to open the scoring two minutes before the break.
The pattern of the first half was largely reproduced during the early minutes of the second and City boss Tony Pulis responded within five minutes by replacing midfielder Salif Diao with striker Ricardo Fuller.
Stoke immediately looked more potent and forced two corners in quick succession, the second of which was headed just wide by defender Ryan Shawcross.
Sensing a change in the balance of the game, Pulis replaced Glenn Whelan with Delap with 56 minutes gone, and his side might have been back on level terms within three minutes.
Jon Walters climbed high above Enrique to return Matthew Etherington's cross towards goal, where Jones headed against the foot of the post with keeper Tim Krul beaten.
The rebound dropped to Fuller, who seemed certain to score, but he stabbed his shot wide of the far post as the home side breathed a huge sigh of relief.
They were counting their blessings again five minutes later when Jones met Dean Whitehead's free-kick with a powerful header, but saw it come back off the crossbar.
But it proved a case of third time lucky with 67 minutes gone when Huth helped Etherington's free-kick back across goal and Jones headed home from point-blank range.
Sorensen claimed Carroll's 70th-minute volley at the second attempt under pressure from Nolan, but the twist in the tale came five minutes from time when Perch powered a header past Krul and into his own net to hand the visitors victory.
Nolan had a chance to claw back a point in injury time, but his right-foot shot flew wide of the far post.
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