The forward, loaned by Stoke from Paris Saint-Germain, struck early in the second half with an impressive finish as part of a lively performance.
Arsenal were aggrieved when Alexandre Lacazette had an equaliser ruled out for offside in a contentious decision.
And that moment proved decisive as the Gunners were unable to repeat their heroics from last week's opener against Leicester City to salvage a result.
The two teams now have three points from their first two matches. Arsenal, who were without the injured Alexis Sanchez, have not won their first two games in a Premier League season since 2009 as they suffer an early blow to their title hopes.
3 - Jesé is the third player to score a goal on his Premier League debut for @stokecity, after Ricardo Fuller & Michael Kightly. Smart.— OptaJoe (@OptaJoe) August 19, 2017
Jese started as one of two Stoke changes and he was the first to threaten after Darren Fletcher's pass played him in, the ex-Real Madrid forward denied when Petr Cech rushed out to save.
Jack Butland was called into action to keep out Danny Welbeck's first-time strike, shortly before the lively Jese led a counterattack that resulted in Eric Maxim Choupo-Moting firing straight at Cech at the other end.
Stoke lost Xherdan Shaqiri to injury midway through the first half, Mark Hughes going with Saido Berahino – dropped from the starting XI after last week's loss to Everton - to replace him as Choupo-Moting, who had started down the middle, moved out wide in a re-shuffle.
Arsenal were dominating possession and Butland had to make a point-blank save after a volley from Aaron Ramsey – in as one of two changes from the Gunners – after Alex Oxlade-Chamberlain's free-kick had set him up, while Stoke captain Ryan Shawcross missed the target with a good headed chance as the match reached half-time level.
But Stoke moved in front within 90 seconds of the restart, Arsenal allowing too much space for Jese to play a neat one-two with Berahino before sweeping a first-time, left-footed finish into the bottom corner.
Arsenal initially responded positively and had a penalty appeal for a Kurt Zouma handball turned down after Ramsey's header across goal had struck the on-loan Chelsea defender.
Butland was unconvincing as he saved from Hector Bellerin, shortly before Welbeck sent a looping header over in space.
But Stoke almost put the result beyond doubt on the break, Berahino's header from Erik Pieters' left-wing cross impressively saved by Cech's outstretched hand.
Arsenal turned to Olivier Giroud from the bench with 24 minutes left, the France striker replacing Sead Kolasinac in a tactical switch and joining Lacazette in attack.
Giroud had an immediate impact, setting up Welbeck, who was denied by Butland from close range.
After Stoke withdrew Jese, there was huge controversy with 18 minutes remaining, Lacazette firing an emphatic finish into the top corner from Giroud's flick, only to be flagged for offside in an extremely close decision.
77.3% - This was the highest possession figure that Arsenal have had in a single Premier League match since 2003-04 (77.3%). Defeated. pic.twitter.com/VnvZYv68Ys— OptaJoe (@OptaJoe) August 19, 2017
Lacazette was replaced along with Granit Xhaka as Arsene Wenger made a last throw of the dice by bringing on Theo Walcott and Alex Iwobi.
Hughes fumed that Shkodran Mustafi went unpunished for a hard late tackle on Pieters, before Giroud headed wide from Nacho Monreal's stoppage-time cross as the Gunners missed their last chance to grab an equaliser.
Key Opta stats:
- Stoke have lost just once in their last eight home Premier League games against the Gunners (W4 D3 L1).
- Stoke City posted their lowest possession figure in a Premier League game (22.7%).
- Indeed, only five teams have won a Premier League game with a lower share of possession than Stoke against Arsenal, since 2003-04.
- Saido Berahino provided an assist for the first time in 57 Premier League appearances, since last assisting in April 2015 against QPR.