The forward gave the hosts a shock lead shortly after the break, and despite late pressure, the Gunners couldn't find an equaliser.
Here's a look at how the two teams lined up:
Stoke had the brighter start, and Jese put Petr Cech to work after just a few minutes. Arsenal struggled to get going, although Hector Bellerin almost picked out Alexandre Lacazette with a good cross before Danny Welbeck went close with a hard shot, but Jack Butland was on hand to make the save.
Bleacher Report's Callum Mackenzie weighed in on the action:
Xherdan Shaqiri lasted just 25 minutes before he was replaced due to an injury, with Saido Berahino coming on.
Aaron Ramsey had a good opportunity before Ryan Shawcross missed a golden opportunity, sending his header over the bar. The Gunners ended the half on top but couldn't threaten the visitors.
Squawka Football shared some half-time stats after a rather dull opening period:
The hosts started the second half with a bang, as Berahino took advantage of some sloppy defending to play in Jese, who finished off the chance for the lead.
ESPN FC's Jonathan Johnson couldn't help himself:
Arsenal tried to hit back with an aerial assault, but Butland displayed tremendous command of his penalty box in dealing with several crosses.
The Gunners wanted a penalty after Kurt Zouma appeared to handle the ball, although replays suggested it was accidental and the youngster didn't have enough time to get out of the way.
Welbeck missed with a header before Berahino almost doubled the lead, drawing a fine save from Cech with a diving header.
Butland again denied Welbeck with a superb save, and when the talented stopper was finally beaten by Lacazette, the official brought the goal back for a controversial offside call.
TV personality and Arsenal fan Piers Morgan was not happy with the decision:
Wenger introduced Alex Iwobi and Theo Walcott in a desperate search for an equaliser, but the Gunners didn't create nearly enough danger late despite their possession.