A goal on the counterattack from Alexis Sanchez looked to have the Gunners back on track before Garry Monk's men riposted with two in three minutes at the Liberty Stadium
After a turgid first half, the Gunners broke through just past the hour when Danny Welbeck and Alexis Sanchez combined for the Chilean to finish off an incisive counterattack.
But two goals in three minutes for the Swans - first a Gylfi Sigurdsson free kick and then a header by Bafetimbi Gomis - turned the match on its head at the Liberty Stadium, leaving the Gunners once again dejected after their midweek mishap against Anderlecht.
Victory for Swansea was only their second in eight Premier League matches and moved them above Arsenal into fifth, with Wenger's men slipping a place further back heading into the international break.
Tottenham loanee Tom Carroll replaced the suspended Jonjo Shelvey in Swansea's midfield, with the returning Leon Britton on the bench to enhance Garry Monk's options, while Kyle Bartley and Marvin Emnes replaced the injured Federico Fernandez and Wayne Routledge.
Arsenal midfielder Mikel Arteta was ruled out due to a hamstring injury so Mathieu Flamini came into the starting line-up.
The visitors were accused of being naive for the manner in which they allowed Anderlecht to fight back on Tuesday and, perhaps as a result, made a cautious start.
Calum Chambers was in the thick of the action, with Wilfried Bony going down in the penalty area, claiming that the England defender had pushed him in the back, after 20 minutes but referee Phil Dowd saw nothing amiss.
Jefferson Montero's pace was causing right-back Chambers problems and the Ecuadorian winger forced a save from Wojciech Szczesny at his near post. The Arsenal goalkeeper was called into action again to deny Emnes after Bony had picked out the former Middlesbrough man.
Lukasz Fabianski was finally called into action a minute before the break against his former club to save from Welbeck after the England international had played a one-two with Alex Oxlade-Chamberlain as Arsenal finished the first half strongly.
Oxlade-Chamberlain had a chance of his own just after the break when Alexis played him in but the midfielder got the ball stuck under his feet before his scuffed shot was easily saved by Fabianski, who was up to the test to deny Santi Cazorla's shot that skidded off the wet surface as rain lashed down.
It came as no surprise when the London club the lead 64 minutes in. Alexis was on hand to apply a simple finish after Welbeck had raced down the right and squared the ball to the red-hot Chile international.
Swansea came roaring back, though, and Sigurdsson gave Szczesny no chance with a moment of class 15 minutes from time as his long-range free kick found the back of the net.
And worse was to come for Arsenal, with Gomis rising to head home Montero's cross soon after replaced Bony to stun the visitors.
Arsenal had a shout for a penalty when they felt that Neil Taylor had handled in stoppage time but referee Dowd waved play on as the visitors suffered only their second top-flight defeat of the season.