The livewire Spaniard, signed on loan from Paris St Germain in midweek, started and finished the decisive move 90 seconds after halftime, slotting home expertly after a sharp link up with substitute Saido Berahino.
In an entertaining game, the former Real Madrid man's goal prompted Arsenal to throw everything at the home defence in a bid to claw their way back but Jack Butland proved a rock in Stoke's tremendous rearguard action.
Arsenal were denied an equaliser when Alexandre Lacazette's strike was ruled fractionally offside but the defeat ensured the Gunners' miserable sequence of being unable to win their first two matches of a season stretched to eight years.
Jese, who had looked threatening even before his goal, was hailed by his manager Mark Hughes.
"Everyone knew what an acquisition Jese is. He's a game-changer. You have to say he put a shift in too," Hughes told the BBC.
"When you think about it, he only had two or three training sessions. If that's what's to come then we'll be delighted. He's an elite player and can carry the fight for you."
For Wenger, though, it was a deflating evening following the optimism of the Gunners' 4-3 win over Leicester City on the opening night of the season.
"I believe some teams will get something here, that is why if you want to play at the top you have to consider it three points dropped," he told BT Sport.
"It was a very frustrating evening. I feel we had plenty of chances to score and didn't. We know in this kind of game it's important not to make a mistake but straight after halftime we were too sluggish on defending and we paid for it."
He also felt new signing Lacazette had been denied his second goal for the club.
"I think we scored a goal that is not offside. It's a very easy decision. When the ball is kicked, Lacazette is not offside," said Wenger.
(Reporting by Ian Chadband, editing by Pritha Sarkar)
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