Danny Graham struck the winner as Swansea claimed the first major scalp of their maiden Premier League campaign.
The Gunners had silenced the home support by taking an early lead through captain Robin van Persie, but Scott Sinclair levelled from the penalty spot for Welsh side.
Nathan Dyer then fired Swansea ahead before Theo Walcott equalised, but Arsenal paid the price for a loss of concentration as Graham ran in to slot his ninth goal of the season less than a minute later to move the Swans up to 10th in the table.
Swansea manager Brendan Rodgers had called on England manager Fabio Capello to visit south Wales to watch his side in action, with Dyer and Sinclair having been linked with a call up to the national squad, and with the Italian in attendance the duo could not have done much more to enhance their aspirations in an enthralling encounter.
But there had been little hint of the drama to come when Arsenal took a fifth-minute lead.
Arsene Wenger had recalled van Persie to his side after his recent extended break, and it did not take the Dutchman long to get back in the swing of things.
Andrey Arshavin's pass carved Swansea apart, and the striker made no mistake as he checked before firing past Steven Caulker and goalkeeper Michel Vorm from a tight angle for his 22nd goal of the season.
The breakthrough initially knocked the wind out of Swansea's sails - but they were level after 16 minutes.
After sharp play down the left by the excellent Joe Allen, Dyer went down under Aaron Ramsey's challenge for referee Michael Oliver to award what looked a soft penalty. Sinclair converted into Wojciech Szczesny's left-hand corner to maintain his 100% record from the spot this term.
The game was living up to predictions of an entertaining contest and the Gunners could have re-established their lead just two minutes later as Ramsey skipped past Vorm and pulled the ball across the six-yard box, only for a fortunate ricochet off Angel Rangel to land in the keeper's arms.
The visitors continued to carve out the clearer chances, and van Persie should have done better when played in just before the half-hour mark.
Swansea captain Ashley Williams was caught hopelessly upfield after Leon Britton had been dispossessed by Yossi Benayoun, the Israeli's pass found his Dutch team-mate in acres of space but Vorm stood up well to repel his right-footed shot.
Arsenal could have mirrored their first half start but failed to make the most of two presentable chances in the opening moments of the second period.
Walcott blazed over after being located down the right channel by van Persie, while Arshavin was equally wasteful after being teed up in space on the edge of the box.
But there was no let such let off at the other end when Dyer put Swansea ahead after 57 minutes.
Ramsey was easily robbed by Wales team-mate Allen and the midfielder picked out Dyer for the winger to shoot high into the net and give Swansea a merited lead.
Wenger had seen enough and Thierry Henry was sprung from the bench along with Tomas Rosicky as the ineffectual Arshavin and Benayoun made way.
Swansea were having their best spell of the game but they were made to pay for a poor lapse in defence as Johan Djourou's pass split Neil Taylor and Williams for Walcott to clip the ball over Vorm and level after 69 minutes.
But the parity lasted less than a minute as substitute Gylfi Sigurdsson's pass saw Graham race in and make the most of Szczesny's hesitancy in advancing from his line to put Swansea back in front.
Per Mertesacker should have levelled barely two minutes later but the German defender horribly scuffed his effort wide, then Ramsey aimed his bullet header from Walcott's cross straight at Vorm as Arsenal threw numbers forward.
The Gunners' last chance came when Rosicky drove to the edge of the Swansea box and forced Vorm to beat out his drive before gathering Laurent Koscielny's follow-up as the home side held out to spark delirious celebrations at the final whistle.