Jese, who was thrust straight into Mark Hughes' side after arriving from Paris St-Germain on Wednesday, slid in from a tight angle after being left unmarked to pick up Saido Berahino's pass.
The Spaniard's strike came just 90 seconds into the second half, after an opening 45 minutes the visitors had dominated.
Arsenal thought they had equalised when Alexandre Lacazette rifled in, only for the France striker to be ruled offside.
The Gunners continued to press in the closing stages but Stoke, who had only 23% possession, remained resolute to earn their first win of the Premier League season.
Can Stoke help Jese rebuild his career?
Stoke have managed to attract a few players with impressive pedigree over recent seasons, but perhaps none more exciting than Spain international Jese.
The 24-year-old forward was considered one of the best products of his generation in the Real Madrid academy by some observers, only for his career to stall after he suffered a serious knee injury - ironically as a result of a challenge from Arsenal opponent Saed Kolasinac, who was then playing for Schalke - shortly before the 2014 World Cup.
After failing to force his way back into Real's plans following his recovery, an underwhelming spell at French giants Paris St-Germain followed before he attempted to rebuild his career with a loan move to hometown club Las Palmas.
The La Liga side decided he did not offer enough to warrant a longer deal - amid reports he was distracted by his rap band - which has allowed Stoke to bring him to the Premier League.
On the evidence of his match-winning debut, it may prove a shrewd signing by Potters boss Hughes.
Jese showed glimpses of his attacking flair and willingness to work throughout, none more so than for what proved to be the only goal of the game.
He started the move on the halfway line, driving forward with pace and purpose, before finding substitute Berahino on the left and then continuing his surge to collect a return pass and convert.
Eventually he ran out of steam - unsurprising after being left out of PSG's pre-season plans - and received a standing ovation when he was replaced after 70 minutes.
Stoke fans will be hoping his goal was the first of many during his season-long loan.
Arsenal's failings exposed again by Stoke
Stoke has long been considered a place where Arsenal's soft underbelly gets exposed against a physical home side - and so it proved again.
Arsenal won for the first time in seven trips to the Potteries when they cruised to a 4-1 victory at the end of last season, leading Arsene Wenger to confidently predict his team could cope with the challenge posed by the home side.
But, as has been an all too familiar sight for Gunners fans, they lacked steel and aggression in key areas of the pitch.
This was illustrated in the build-up to Stoke's winner.
A sloppy pass from Granit Xhaka towards a flat-footed Mesut Ozil gave the ball back to Stoke, with neither player working hard enough to retrieve the ball as Jese broke forward.
"We know not to make defensive mistakes and we came out after half-time too sluggish. We were punished," said a frustrated Wenger.
This graphic shows the ease with which Stoke netted their winner. Geoff Cameron (20) beat Ozil to Xhaka's stray pass, feeding Jese (11) on the halfway line before he did the rest with the help of Berahino (9)
In truth, Arsenal could have already been well in command after a dominant first-half performance in which they had over 70% possession.
Still without the injured Alexis Sanchez, the visitors forced Potters keeper Jack Butland into a string of a splendid saves, particularly from angled drives by Aaron Ramsey and Danny Welbeck, while Arsenal's players also felt they should have been given a penalty when Hector Bellerin was clumsily bundled over by Stoke forward Mame Biram Diouf.
Wenger was also aggrieved that Lacazette was called offside by the tiniest of margins as they chased an equaliser.
"I believe we scored a regular goal that was not offside. It is an easy decision. When the ball is kicked he is not offside," he added.
The Gunners boss threw on substitutes Oliver Giroud - who rescued them in their opening win against Leicester - Theo Walcott and Alex Iwobi in the final stages, but the forwards could not mask their team's defensive deficiencies for the second game running.
Man of the match - Jack Butland (Stoke City)
Picking the keeper from a winning side might seem a slightly odd choice, but Stoke might not have won had England international Butland not made three superb saves before half-time. Three more stops after the break ensured the Potters clinched their first win of the season.
Possession is not everything... - the stats
- Stoke have lost just once in their past eight home Premier League games against the Gunners, winning four and drawing three.
- Arsenal posted their highest amount of possession in a Premier League game (77.3%) since 2003-04.
- Only five teams have won a Premier League game with a lower share of possession than Stoke.
- Since Stoke City's first Premier League campaign in 2008-09, Arsenal have suffered six defeats away at the Potters in all competitions. In this period, they've only suffered more competitive defeats at Manchester United (eight) and Chelsea (seven), and have also lost six times at Manchester City.
- The Gunners (732) attempted 520 more passes than Stoke (212) in the game.
- Arsenal have registered just one clean sheet in their past nine top-flight away games.
- Arsenal have now failed to win both of their opening two league games every season since 2009-10.
- Jese is the third player to score on their Premier League debut for Stoke, after Ricardo Fuller in 2008 and Michael Kightly in 2012.
- Berahino provided an assist for the first time in 57 Premier League appearances, since last assisting in April 2015 for West Brom against QPR.