A horrendous error from goalkeeper Petr Cech allowed Jordan Ayew to score a tap-in which put the hosts ahead just after the hour mark, before Sam Clucas' second goal made the result safe.
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Mesut Ozil's sensational pass had allowed Nacho Monreal to open the scoring for the Gunners, but Clucas levelled for Swansea within a minute at the Liberty Stadium.
Swansea earned back-to-back wins for the first time this season, while Arsene Wenger's side have won only three of 13 Premier League away games this term.
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