The former Barcelona forward sent Lukasz Fabianski the wrong way after he was bundled over by Ashley Williams.
Stoke, now 11th in the table, had chances to extend their lead through Glen Johnson, Charlie Adam and Joselu.
Jonjo Shelvey went close to equalising when his low, angled shot hit the inside of a post and bounced out.
Yet this was a distinctly under-par performance by Garry Monk's side, who drop to 14th in the table and are without a win in five league games.
Stoke were creative, industrious, dogged and determined as they dominated the opening 45 minutes.
The hosts rallied after the break, substitute Gylfi Sigurdsson going close with a dipping shot soon after Shelvey had been denied by the woodwork.
Worrying times for Monk
Monk deservedly won plenty of plaudits for leading Swansea to an impressive eighth-placed finish last season.That was partly down to ruthless home form, with Arsenal and Manchester United among nine teams who left the Liberty Stadium without any points.
But these are worrying times for Monk and his players. The Swans look desperately short of confidence after a return of just two points from a possible 15.
Monk needs to get the Swans, who have an awkward trip to fellow strugglers Aston Villa on Saturday, firing quickly again.
"We went five games without a win last season. I'm not worried. We'll come through it no problem," said the Swansea boss afterwards.
Stoke's rapid rise
Stoke were left propping up the Premier League table with two points from five games after a 2-0 defeat at Arsenal on 12 September.Yet Mark Hughes' side have moved above champions Chelsea and are only two points behind seventh-placed Tottenham after an impressive change in form.
They bossed Swansea from the start, with Williams' foul on Bojan a gift which cost their Welsh opponents dear.
Stoke have been guilty this season - against Norwich and Leicester - of failing to hang on to a lead.
Against Swansea, there was little doubt of the outcome once Bojan scored.
However, there was a moment of concern when keeper Jack Butland came charging out and committed a studs-up challenge on Ayew which went unpunished.
The incident was replayed on the big screen inside the ground provoking a chorus of jeers from Swans fans.
"I was a little bit upset that the replay was shown in the stadium," said Hughes. "That shouldn't be allowed. It isn't allowed."
Man of the Match - Joselu
What they said
Swansea City manager Garry Monk: "We found it difficult to break them down. I think we deserved a point but I can't grumble with a defeat."We've had a month of not hitting those levels.
"They're very good players. I need to get them back to that level. I have big confidence in this group."
"Maybe if we'd taken more care with the final pass we might have capitalised on more opportunities.
"They never really created anything of note. We've restricted them and created opportunities.
"Swansea never got any momentum in the game which was important. We're on a decent run of form."
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